Tomasz Zaremba

Huw Thomas

Michael Cole

Elizabeth R Plummer

Nicola J Curtin
email: n.j.curtin@ncl.ac.uk

Doxorubicin-induced suppression of poly(ADP-ribose

Keywords: PARP-1, PARP-1 activity, PARP-1 expression, PARP inhibitors, Doxorubicin, Monozygotic twins

Introduction and aims

Monozygotic twins, due to identical genetic constitution, uterine and early environment, provide an excellent material to study the contribution of genes and the environment on human biochemistry, pharmacology and health. Twin studies can help to identify underlying causes of both physical and mental disorders [START_REF] Boomsma | Classical twin studies and beyond[END_REF] and response to drugs [START_REF] Ring | Gene-gene interactions between CYP2B6 and CYP2A6 in nicotine metabolism[END_REF]. Large, nationwide twin studies can help to deWne the contributions of nature and nurture to complex and multifactorial diseases such as cancer [START_REF] Pedersen | The Swedish Twin Registry in the third millennium[END_REF]. The identiWcation of areas of study likely to be worth investing the substantial resources needed for the larger studies comes from initial interesting sporadic observations. Here, for the Wrst time, we investigated PARP activity and expression in monozygotic twin brothers excluding any potential genetic factors that could aVect levels of measured parameters. As a part of a pharmacogenetics study, the elderly male twins, one with a 123 sarcoma that was treated with doxorubicin (DOX) and the other healthy, were recruited to the study.

Soft tissue sarcomas are rare types of cancer accounting for about 1% of adult malignancies (http://www.macmillan. org.uk/Cancerinformation/Cancertypes/Softtissuesarcomas/). Sarcomas are usually treated by a surgical excision either alone or in combination with radiotherapy. There are only three drugs, DOX, dacarbazine and ifosfamide, currently used in treatment that give a relatively good response rate of 20% or more in patients with advanced soft tissue sarcomas [START_REF] Cormier | Soft tissue sarcomas[END_REF].

DOX (also called adriamycin) is one of the most widely used anticancer agents and has the broadest spectrum of activity involving DNA intercalation, free-radical formation, covalent DNA binding and inhibition of the topoisomerase II resulting in DNA strand breakage. DOX is a widely prescribed chemotherapeutic used to treat many diVerent types of cancers; however, its use in high doses is limited by the occurrence of severe side eVects such as cardiotoxicity, myelotoxicity and Palmar-plantar erythrodysesthesia [START_REF] Demetri | Major developments in the understanding and treatment of soft tissue sarcomas in adults[END_REF].

Poly(ADP-ribose) polymerase-1 (PARP-1) is a DNAbinding enzyme activated by DNA single-strand and double-strand breaks and involved in DNA repair, genomic stability, transcription control, cell death and proliferation. Inhibition, or lack, of PARP-1 results in hyper-sensitivity to DNA methylating agents, topoisomerase I poisons and ionising radiation, all used in the treatment of cancer [START_REF] Curtin | PARP inhibitors for cancer therapy[END_REF]. Recent in vivo studies demonstrate that PARP inhibition potentiates the antitumour eVects of DOX in p53-deWcient human breast cancer [START_REF] Mason | INO-1001, a novel inhibitor of poly(ADP-ribose) polymerase, enhances tumor response to doxorubicin[END_REF].

Analysis of the samples received from the twin subjects allowed us to investigate the inter-individual variation in PARP activity in subjects with the same genetic background, and we could also investigate how DOX aVects PARP activity in peripheral blood mononuclear cells (PBMCs). The observed decrease in PARP activity following DOX treatment in the patient was then studied in vivo in mice treated with DOX.

Materials and methods

All chemicals and reagents were of the highest quality and supplied by Sigma (Dorset, UK), unless indicated otherwise. The PARP inhibitor AG014699 was supplied by PWzer GRD. The 10H mouse monoclonal primary antibody was generously supplied by Prof. Alexander Bürkle (University of Konstanz, Germany).

Human subjects

The PARP pharmacogenetics study has been reviewed by the appropriate ethics committee and has been performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki. The patient and his healthy twin brother (control subject) were consented to donate blood samples under the regulations of local ethics committee. A volume of 20 ml or less of whole blood was collected from the subjects by venipuncture. Blood from the patient was collected just before the Wrst administration of chemotherapy and 21 days after the Wrst administration, just before second cycle of treatment. This was transferred into EDTA blood tubes (BD Vacutainer, Plymouth, UK). PBMCs were then extracted from blood using Lymphoprep™ Tube (Axis-Shield, Norway) according to manufacturer's instructions. The cell pellet was resuspended in 500 l prechilled freezing medium (RPMI plus 10% foetal bovine serum, FBS), supplemented with 10% DMSO, and 1£ Halt protease inhibitor cocktail (Thermo Fisher ScientiWc, Rockford, IL). Samples were stored immediately at ¡80°C.

Animals

The experiments on animals were conducted in accordance with national law and institutional guidelines for the care and use of animals under a protocol approved by local ethics committee. Male BALB/C mice (8-10 weeks of age, 5 mice/group) were treated as follows: untreated controls and mice treated with 5 mg/kg DOX via intraperitoneal injection (IP) and killed 1 h, 24 h and 6 days after treatment. Blood from the Wve animals was pooled to yield suYcient sample for lymphopreparation and collected into tubes containing an anticoagulant (EDTA) and stored on ice for up to 2 h prior to further processing. Blood was then diluted 1:1 with phosphate buVered saline (PBS) 6 ml was layered over 3 ml of Lymhoprep (Axis-Shield, Norway) and processed according to manufacturer's instruction.

Measurement of PARP activity

PARP activity was measured by modiWcation of a previously described method [START_REF] Plummer | Temozolomide pharmacodynamics in patients with metastatic melanoma: dna damage and activity of repair enzymes O-6-alkylguanine alkyltransferase and poly(ADP-ribose) polymerase-1[END_REF] validated to GCLP standard and used as a pharmacodynamic endpoint for clinical trials [START_REF] Plummer | Phase I study of the poly(ADP-ribose) polymerase inhibitor, AG014699, in combination with temozolomide in patients with advanced solid tumors[END_REF]. BrieXy, maximally stimulated PARP activity was measured in triplicate samples of 10 4 cells permeabilised with icecold digitonin (0.15 mg/ml in water) in a reaction mixture containing 350 mol/l NAD + as substrate in excess and 10 g/ml PARP-activating oligonucleotide (CGGAATT CCG) (Europrim, Invitrogen, UK) that mimics DNA strand breaks, in a reaction buVer of 100 mmol/l Tris-HCl and 120 mmol/l MgCl 2 (pH 7.8) in a Wnal volume of 100 l at 26°C in an oscillating water bath. The use of an oligonucleotide to maximally stimulate PARP minimises variation due to accidental activation of PARP by the unintentional induction of DNA breaks during PBMC harvesting and 123 processing. The reaction was stopped after 6 min. by the addition of excess PARP inhibitor (400 l of 12.5 mol/l AG014699), and the cells were blotted along with a poly(ADP-ribose) standard (0-25 pmol: Biomol, Exeter, UK) onto a nitrocellulose membrane (Hybond-N, Amersham, UK) using a purpose-built 48-well manifold. Following incubation with the primary anti-PAR 10H antibody 1:500 in PBS containing 5% non-fat powdered dried milk and 0.5% Tween 20 (PBS-MT) overnight at 4°C then HRP-conjugated goat anti-mouse secondary antibody (1:1,000 in PBS-MT; Dako, Ely, UK) for 1 h at room temperature, the membrane was exposed for 1 min. to enhanced chemiluminescence (ECL) reaction solution (Amersham, Little Chalford, UK), and chemiluminescence was measured using a Fuji LAS3000 with imaging software (Fuji LAS Image version 1.1, Raytek). Results were expressed in light absorbtion units (LAU) relative to the number of cells loaded and subsequently calculated by reference to the poly(ADPribose) standard curve.

Measurement of PARP-1 expression

BrieXy, the cell pellet was lysed in 100 l Laemmli buVer [START_REF] Laemmli | Cleavage of structural proteins during the assembly of the head of bacteriophage T4[END_REF] with 1£ Halt protease inhibitor cocktail (Thermo Fisher ScientiWc) resuspended on ice for 30 min. sonicated on ice for 10 s (Vibracell Sonicator, Sonics and Materials, Danbury, Conn.) and heated in loading dye containing -mercaptoethanol and bromophenol blue at 95°C for 5 min. Lysates (30 g of protein per lane) were loaded onto Tris-HCl 5-20% polyacrylamide gels (Bio-Rad, Hercules, CA) along with puriWed recombinant PARP-1 immunoblotting standard (Biomol, Exeter, UK), and after electrophoresis at 100 V for 2 h (Criterion electrophoresis apparatus, Bio-Rad), the proteins were transferred for 1 h at 4°C into a nitrocellulose membrane (Hybond-C, Amersham, UK) using a Criterion Blotter (Bio-Rad). After blocking for 1 h in PBS-MT, the membrane was incubated overnight at 4°C with shaking with an anti-PARP-1 C2-10 primary antibody (Trevigen, Gaithersburg, MD) diluted 1:2,000 in PBS-MT, washed 3 times for 15 min. in PBS containing 0.5% Tween 20 (PBS-T) and then incubated with the HRP-linked secondary goat anti-mouse antibody (Dako) diluted 1:1,000 in PBS-MT, washed again for 1 h in PBS-T with changing buVer every 5 min. and then dried. The protein was visualised with ECL plus detection kit (Amersham) using the manufacturer's protocol followed by chemiluminescence detection as described above. We quantiWed the PARP-1 expression by reference to puriWed recombinant PARP-1 protein (Biomol) standard curve (0-40 ng). This assay has also been validated to GCLP standard for evaluation of patient samples (E. Mulligan and T. Zaremba, unpublished data).

Statistical analysis

We analysed each pooled sample in triplicate in three independent experiments (animals) or in duplicate (Western blot) and quadruplicate (PAR assay) in single experiment (humans). We calculated P values for PARP activity comparison by the Student's t test using GraphPad Prism4 software (GraphPad, La Jolla, CA) and by Tukey's test (Minitab, Coventry, UK).

Results

Case presentation

A 67-year-old male patient presented in 2006 with a 7-8 cm left proximal vastus medialis mass in the left thigh mass, which on incision biopsy showed a Trojani grade 3 soft tissue sarcoma of non-speciWc type, and no evidence of metastatic disease. Subsequently, the patient had surgery to remove the tumour and received adjuvant radiotherapy (60 Gy in 30 fractions). In 2007, pulmonary metastases and intra-abdominal lymphadenopathy were detected, and the patient received single agent DOX (142.5 mg/m 2 ). Before the Wrst cycle of chemotherapy, the patient was consented for study. A blood sample was obtained and processed according to local ethics committee. The patient revealed that he had an identical twin brother who would be willing to donate a blood sample for the study as a healthy volunteer. The sample was obtained at the same time as the second sample from patient, which took place 3 weeks after administration of the Wrst dose of DOX.

PARP activity

Measurement of PARP activity in PBMCs from the subjects showed that the mean PARP activity for the patient before the treatment was 160 pmol PAR/10 6 cells and that for the healthy volunteer was 130 pmol PAR/10 6 cells. We observed an approximately ninefold decrease in PARP activity in a second sample from the patient taken 21 days after the Wrst drug administration (17 pmol PAR/10 6 cells) (Fig. 1a,b).

PARP-1 expression

Measurement of PARP-1 expression in PBMCs from the subjects showed that the mean expression for the patient before the treatment was 0.15 ng PARP/ g total cellular protein while for the healthy volunteer was 0.2 ng PARP/ g total cellular protein. We observed a decrease in PARP-1 expression to below the limit of quantiWcation in the patient sample taken after the Wrst cycle of DOX treatment (Fig. 1c).

PARP activity in mice treated with DOX

The eVects of DOX treatment (5 mg/kg) on PARP activity in PBMCs from mice 1 h, 24 h and 1 week after drug administration are shown in Fig. 2. We observed decrease in PARP activity 1 h after drug administration (63% control, range 50-88%; P ¿ 0.05, Tukey's test) and after 24 h (53% control, range 12-78%; P ¿ 0.05, Tukey's test). We observed restoration in PARP activity 1 week after DOX treatment (86% control, range 66-112%; P = 0.24, Tukey's test).

Discussion

In this study, we investigated the inter-individual variation in PARP activity and expression in subjects with the same genetic background i.e. homozygotic twins. We found that in the patient's pre-treatment sample, PARP activity and expression were similar to his identical brother. In both subjects, PARP activity was within the range seen in the general population of healthy volunteers and cancer patients (10-2190 pmol/10 6 cells: TZ unpublished data). We showed that an identical genotype gave very similar phenotype, and this Wnding must be taken into account in an ongoing discussion about the factors determining PARP activity in relation to PARP-1 expression in humans. We also found that after DOX treatment, the patient had much lower PARP activity and PARP-1 expression. This observation suggested that DOX treatment may reduce both parameters.

To conWrm our Wndings, we also investigated DOX treatment in an animal model: PARP activity was also suppressed in male mice treated with DOX (5 mg/kg) 1 and 24 h after drug administration (P ¿ 0.05). However, the eVect was more modest (approximately twofold) compared to the substantial (ninefold) reduction in the patient.

Our Wnding with a decrease in PARP activity after DOX treatment is somewhat in contrast with previous data showing that cardiotoxicity of DOX is a result of oxidative stress (reactive oxygen species, ROS formation) where oxidative DNA breakage leads to PARP activation [START_REF] Pacher | Activation of poly(ADP-ribose) polymerase contributes to development of doxorubicin-induced heart failure[END_REF]. We hypothesise that DOX treatment, due to its selective toxicity to diVerent subtypes of blood cells, altered the proWle of the PBMC (mostly lymphocytes) population and that the observed decrease in PARP activity could be due to selective killing of the population of cells with higher PARP activity by DOX. Ranjit et al. [12] showed that diVerent subtypes of PBMCs had diVerent PARP-1 content with B lymphocytes (18-47% of lymphocytes) having the highest expression level (8.64 Units), compared with T lymphocytes (approx. 70% of lymphocytes, 4.2 Units) and monocytes (3-8% of leucocytes, 2.2 Units). If the reason for high PARP-1 expression in B lymphocytes is associated with its role in DNA rearrangement necessary for the production of antibodies, PARP activity in B cells may also be high. At least two studies showed that chemotherapy causes signiWcant reduction in B lymphocyte count. Mackall et al.

[13] demonstrated a rapid and severe decrease in B cell number to almost undetectable level after one cycle of chemotherapy (several diVerent regimes including single-agent cyclophosphamide and cyclophosphamide in combination with DOX and vincristine) and up to sixfold decrease in number of T cells in 10 studied patients. Similarly, Hakim et al.

[14] showed up to 96% reduction in B cells and >60% reduction in T cells after treatment with FLAC regime (DOX, 5-Xuorouracil, cyclophosphamide). The full recovery in number of B lymphocytes was not observed until 3 months after the last cycle. These data suggest that chemotherapy leads to the depletion of several subtypes of PBMCs, especially B lymphocytes. The decrease in activity in the sarcoma patient treated with DOX could therefore be due to loss of PBMCs with high PARP-1 expression and PARP activity leaving the patient with those that naturally have less PARP-1 and lower activity. However, a larger study would be needed to conWrm such an implication. The much greater eVect of DOX in the patient than in the mice may reXect the time of sampling. If the high PARP-1expressing cells are culled in a time-dependent fashion, we may not have sampled the mouse PBMCs at the nadir and may have been only measuring the semi-acute eVects of DOX on PARP activity rather than a combined eVect on activity and selective killing of high PARP-1-expressing cells. Further experiments are clearly warranted to determine the time-dependent eVects of DOX on lymphocyte populations with diVerent PARP-1 expression. This small study showed a profound decrease in PARP activity measured in PBMCs from a patient who underwent chemotherapy treatment. Our hypothesis that this could be due to a selective depletion of B lymphocytes in blood requires further investigation. This Wnding is very important for current and future clinical studies and trials on PARP inhibitors where PARP activity is used as a pharmacodynamic endpoint; here, we demonstrated that a decrease in PARP activity in such studies could be due to an eVect of the chemotherapeutic, as well as the PARP inhibitor, on PBMCs and lead to a false impression of the eVectiveness of the PARP inhibitor. As a potential solution it would be beneWcial to perform Xow cytometry analysis to test the content of diVerent cell subtypes and subsequently measure PARP activity in those cells. The diVerence in PARP activity before and after drug administration would give a real picture of eVectiveness of tested therapy and also information on its toxicity to particular cell subtypes.

Fig. 1 Fig. 2

 12 Fig. 1 PARP activity and expression measured in PBMCs from the patient before and after Wrst cycle of DOX treatment and from the healthy volunteer (control subject). a PAR formation detected by immunoblot assay. First row of dots shows results for 25, 10, 5, 1, 0.2, 0 pmol PAR standard and positive reaction for quality control sample of L1210 cells (positive control). Dots in rows I-II show PAR formation in PBMCs from the patient before and after DOX treatment, row III shows PAR formation in PBMCs from healthy subject. b Semiquantitative analysis of the results of the experiment shown in a: mean

Acknowledgments We thank Prof. Alexander Bürkle for 10H antibodies and PWzer GRD for AG014699 inhibitor. TZ was supported by a studentship from AICR, and NJC is supported by Cancer Research UK.

ConXict of interest statement

None.


Kristina Giesel

Johannes Tambornino

Thomas Thiemann

LTB spacetimes in terms of Dirac observables

come

Introduction

Until today General Relativity (GR) is the most successful theory in describing observations involving gravitational interactions on macroscopic scales. It has been tested in various experimental settings and so far many predictions of GR have been confirmed to great accuracy. But next to the complicated highly non -linear structure of Einstein's equations there is one aspect in GR which has to be treated with special care as compared to other field theories such as Maxwell's theory: the issue of observables. Observables are those quantities that respect the gauge symmetry of the theory, for Electrodynamics this would simply be all U (1) -invariant quantities. In contrast for GR the gauge group is closely related to the group of space time diffeomorphisms, denoted by Diff(M ), reflecting the independence of physics on space time coordinates. We are dealing with a background independent theory where "space" and "time" do not have any a priori physical meaning. In the canonical picture of GR [1], where one performs a (3+1) -split of the four dimensional space time (M, g), gauge invariance carries over to the condition that phase space functions have to commute with the Hamiltonian and spatial diffeomorphism constraints of GR1 . The reason why it is so complicated to extract the gauge-invariant content of GR is that the Poisson algebra formed by the constraints of GR, the so called Dirac algebra, is an extremely difficult one. It is not only non-Abelian but is not even an honest Lie-algebra: instead of structure constants it involves structure functions, i.e., phase space dependent quantities. However, even if observables were known there would still be a conceptual task to solve which is often referred to as the problem of time in GR [4,5] and concerns the evolution of observables. Speaking about dynamics of GR we usually refer to Einstein's equations. In the canonical picture these can be obtained from a canonical Hamiltonian and the Hamiltonian and diffeomorphism constraint equations. However, as a consequence of diffeomorphism invariance the canonical Hamiltonian is a linear combination of constraints only2 . Hence, the Hamiltonian equations generated by this canonical Hamiltonian will be identical to zero in the case of observables. Consequently, Einstein's equations do not describe what we usually would call physical evolution but rather describe how the metric changes under gauge transformations. What is needed for observables is a gauge invariant version of Einstein's equation, generated by a so called physical Hamiltonian and describing non -trivial evolution of observables. A framework that allows to construct observables and analyse their evolution for constrained systems such as GR is the so called Relational Formalism. The basic idea is to take the background independent nature of GR seriously and define observables not with respect to unphysical space time points in M but to use relations between dynamical fields instead. These ideas date back to the seminal work of Bergmann and Komar [6,7,8] from the 1960s. Its conceptual foundations were very much improved in the 1990s, see [9,10] and references therein. The corresponding mathematical framework was also developed in the 1990's, see e.g. [11] and references therein and rediscovered more recently in [12,13]. Once appropriate dynamical fields are chosen as clocks3 in order to give space and time a physical meaning one can at least formally write down an expression for observables associated to any phase space function f . In [14,15] a perturbative scheme to compute these observables was developed and its application to perturbation theory around Minkowski space and cosmological perturbation theory for different choices of clocks was discussed. However, the explicit form is a power series in these clock variables with coefficients involving multiple Poisson brackets of the constraints and f generally leading to a rather complicated dependence on the physical time parameter that in most cases cannot be written down in a closed analytic form.

However, as pointed out for instance in [16], the precise expression of the Dirac observables in terms of non gauge inavriant objects, which generically involves a hopelessly complicated, infinite series of multiple Poisson brackets, is in fact irrelevant from the point of view of the physical (or reduced) phase space that we are actually interested in. What is relevant is the Poisson algebra of those Dirac observables and the physical Hamiltonian which drives their physical time evolution. General expressions for an arbitary first class system and for general choices of clocks were derived in [16]. From a practical point of view and especially with regard to quantisation it is of course desirable to have a sufficiently simple gauge invariant Poisson algebra A phys and a compact expression for the resulting physical Hamiltonian H phys . The structure of both A phys , H phys depends sensitively on the choice of clocks and for generic choices both are beyond mathematical control. One therefore should analyse first which type of matter simplifies their structure.

First steps in this direction have been performed in [17] where a phantom scalar field, i.e. a scalar field with negative kinetic energy term, has been used as a clock and the corresponding physical Hamiltonian has been derived. The choice of a scalar field with vanishing potential as a clock field was motivated entirely by mathematical considerations based on the Brown -Kuchař mechanism [18] which allows to deparametrise the Hamiltonian constraint. From a physical point of view, introducing some ad hoc matter component that adopts the role of clocks for GR might look artificial and even dangerous, especially if that matter is in conflict with the ususal energy conditions that stabilise the system. However, the way that the relational formalism works is in fact very similar to the Higgs mechanism: In the gauge invariant formulation, the scalar field completely disappears. What remains is the corresponding physical Goldstone boson and what one has to worry about is that this additional degree of freedom is not in conflict with observation, in particular, that the energy conditions hold in the gauge invariant description.

A more physically motivated choice of clocks which have the additional advantage of implementing the Brown -Kuchař mechanism is defined by the original Lagrangian for pressure free dust due to Brown and Kuchař [18]. This choice of matter is a sense physically distinguished because it can be considered in a precise sense as a congruence of mutually non interacting, freely falling observers which only interact gravitationally and which define the dynamical reference frame for GR. From the mathematical point of view, the essential point in the construction is that the constraints of the coupled system, including gravity, matter and dust, can be written in a deparametrised form4 . As was shown explicitly in [19], this achieves the goal of drastically simplifying the Poisson algebra of physical observables and the physical Hamiltonian and furthermore ensures that the associated physical Hamiltonian is time independent. In contrast to [18], the requirement that the physical Hamiltonian should be positive definite, or equivalently physical time is required to run forward rather than backwards, demands the dust needs to be phantom dust. This is exactly the same reason why the phantom occurred in the previous work [17] and ensures that the energy momentum tensor of observable matter obeys the usual energy conditions. It was also shown that the resulting theory is in good agreement with current cosmological observations in [20] as far as FRW space times as well as perturbations around FRW are considered. The reason for why that happens is the afore mentioned Higgs mechanism by which the phantom dust completely disappears from the physical particle spectrum together with another nice feature of this particular choice of clocks which is absent for generic choices: The physical Hamiltonian system posesses an infinite number of conserved charges which can be considered as physical energy and momentum densities of the dust respectively and which enforce the corresponding gravitational Goldstone boson modes to decouple in a mathematically precise sense from the physical time evolution. In particular, it is consistent to tune those charges to be arbitarily small so that their corrections to Einstein's equations for the non Goldstone modes are also arbitrarily small. In this way the dust comes as close as possible to the mathematical idealisation of a test observer while taking its gravitational interaction into account. Of course, one could also use gravitational clocks, that is certain components of the metric tensor and thus avoid the Goldstone modes altogether. However, while that works well for the linearised theory, taking into account the full non linearities of Einstein's theory leads to equations of motion for the physical graviational modes that have no resemblance with the classical Einstein equations whatsoever. Similar to the vast number of proposals from cosmologists, particle physicists and string theorists for possible extensions of the field content of the standard model such as inflatons, axions, dark matter, supersymmetric extensions, dilatons, Kaluza Klein modes etc. we consider the existence of a fundamental dust field in addition to standard matter as an at least theoretical possibility which of course has to be tested in experiments. The fact that the dust only interacts gravitationally makes it in principle a perfect dark matter candidate 5 .

The topic addressed in this article is the spherically symmetric sector of the theory presented in [19]. We will show in detail that a gauge invariant version of Einstein's equations specialised to the spherically symmetric case can be mapped to a family of Lemaitre -Tolman -Bondi (LTB) solutions. The only modification is that the equations contain a phantom dust energy density instead of the usual dust energy density and we will discuss explicitly which consequences this has.

As pointed out in [16], the existence of gauge invariant version of Einstein's equations and thus a reduced phase space formulation of GR involving an algebra of observables and a physical Hamiltonian is of advantage when a quantisation of gravity is concerned because one sidesteps the difficulties involved in the anomaly free quantisation of constraints and the construction of the physical Hilbert spaces. Recently, this strategy has been used in order to present a reduced phase space quantisation [21] for Loop Quantum Gravity [22,23]. This framework developed for the full theory can of course be applied also to the spherically symmetric sector [24].

The paper is structured as follows:

In section 2 we review the results of [19] for the full theory. In section 3 we specialise the observables and the physical Hamiltonian to spherically symmetric space times and in section 4 derive the corresponding equations of motion. These equations are a gauge invariant version of Einstein's equations in the case of spherical symmetry. Section 5 discusses the solution of these equations which belong to a family of LTB -solutions. In section 6 and 7 we discuss further properties of these solutions such as their semistatical behaviour as well as the occurrence of singularities. For the latter we analyse in particular the effect of the phantom dust energy momentum that occurs as a source in these equations. Finally in section 8 we conclude, discuss the implications of the work done in this paper and give an outlook. Several appendices on the boundary conditions imposed on gauge invariant variables, various spherically symmetric coordinate systems employed in the literature and a comparison with the covariant derivation of the LTB solutions complete the paper.

Brown -Kuchař Dust Reduction of General Relativity

In this section we summarise the analysis performed in [19] which builds on the seminal work [18].

The Brown -Kuchař dust Lagrangian is given by

S dust = - 1 2 M d 4 X | det(g)|ρ[g µν U µ U ν + 1] ,
(2.1)

5 One could call it a NIMP (Non Interacting Massless Particle) as compared to a WIMP (Weakly Interacting Massive Particle) which is considered as the most favourite (cold) dark matter candidate. Massless here refers to the fact that the dust Lagrangian does not contain a usual mass (or potential) term. However, the foliation defined by the dust is always spacelike and the corresponding foliation vector field has unit timelike norm with respect to the physical metric (but it is not necessarily normal to the foliation). That is, the dust moves along unit timelike geodesics which also follows from the fact that the dust Lagrangian can be interpreted as the sum (or integral) of relativistic point particle Lagrangians (one for each flow line of the congruence) with variable mass depending on the flow line. This mass distribution is just the energy density of the dust. See [18] for details. Hence the dust moves at non relativistic speeds as required by realistic dark matter models. Of course the dust energy density has the wrong sign to explain the anomalous galactic rotation curves but we should recall that the dust is anyway like a Higgs boson which disappears from the observable particle spectrum. There is no obstacle in adding one's favourite observable dark matter model.

which is coupled to the standard Einstein Hilbert action S EH

S EH = 1 κ M d 4 X | det(g)| (4)
R.

(2.2)

Standard matter Lagrangians can be added in the usual way, but they do not couple to the dust. Here M is a four dimensional manifold which can topologically be identified with R × X for some three dimensional manifold X of arbitrary topology and X are local coordinates on M . Next, g µν (X) with µ, ν = 0, 1, 2, 3 denotes a (Pseudo-) Riemannian metric on M . U ∈ T * M is a one form defined as the differential U = -dT +W j dS j , j = 1, 2, 3, for some scalar fields T, W j , S j ∈ C ∞ (M ). So finally the action (2.1) is a functional of g µν and eight scalar fields ρ, T, W i , S i . (4) R is the Ricci scalar corresponding to the metric g. We need the Hamiltonian formulation of that system which can be derived by performing a usual ADM -(3+1) -split with respect to a foliation of M ∼ = R × X . We denote the momentum conjugate to the ADM 3 -metric q ab by p ab . The momenta conjugate to the configuration variables ρ, W j , T, S j are denoted by Z, Z j , P, P j respectively. Latin indices range from 1 to 3. A detailed Dirac analysis of the occurring constraints shows that the coupled system is second class. Hence, in order to proceed, one passes on to the corresponding Dirac bracket and solves the second class constraints explicitly. As a consequence it turns out that the momenta conjugate to ρ and W j vanish and that these configuration variables can be expressed in terms of the remaining phase space variables. Explicitly, we have

Z := 0, Z j := 0, W j := - P j P , ρ 2 := P 2 √ det q q ab U a U b + 1 . (2.3)
In general, analysing second class constraint systems is a very hard task due to the complicated structure of the corresponding Dirac bracket. However, it turns out that for the system S EH + S dust , when restricting attention to the geometry variables (q ab , p ab ) as well as the remaining dust variables (S j , P j ) and (T, P ), then the Dirac bracket in this sector reduces to the standard Poisson bracket again, which makes a further analysis of this system tractable.

We end up with a first class system possessing the following constraints

c tot = c + c dust (2.4) c tot a = c a + c dust a , (2.5)
where the geometry and dust contributions can be written as

c a := - 2 κ q ac D b p bc (2.6) c := 1 κ 1 √
det q q ac q bd -1 2 q ab q cd p ab p cd -det(q)R(q) (2.7)

c dust := P 1 + q ab U a U b (2.8)
c dust a := P T ,a + P j S j ,a .

(2.9)

Here D a is the covariant differential compatible with q ab and R(q) denotes the Ricci scalar of q ab . Here P takes only non positive values and thus the energy density of the dust is negative or zero which is why we call it phantom dust in contrast to [18]. The reasons for that can be summarised as follows: If P would take positive values then on the constraint surface we would have c < 0. As we will see, the derived physical Hamiltonian is approximated by |c| = -c which in the limit of flat space would be the negative of the standard model Hamiltonian. One could cure this by letting time run backwards and defining the physical Hamiltonian by -|c| = c but then energy would be unbounded from below. Furthermore, the dynamical foliation generated by the dust would be past oriented if we chose the other sign, see [19] for a detailed discussion.

Concerning the interpretation of (2.1) we refer the reader to [18], where the authors describe that the dust action can actually be derived as a field theoretic generalisation of the concept of free massive relativistic particles moving on geodesics of the gravitational field created by the entire collection of particles. To get an idea why this interpretation is possible, one can check that the integral curves of U µ = g µν U ν describe geodesics of g µν , S j is constant along each of these curves and T describes proper time. So S j = const. labels a geodesic and T = const. is an affine parameter along the geodesic. This is exactly the reason why it so convenient to use the fields T, S j as a physical reference frame.

The canonical Hamiltonian that generates Hamiltonian equations for (q ab , p ab ), (P, T ) which are, together with the constraints in (2.4) and (2.5), equivalent to the 10 Einstein equations obtained in the Lagrangian framework is given by

H can := X d 3 x n(x)c(x) + n a (x)c a (x) .
(2.10)

where n is the so called lapse function and n a the so called shift vector which play the role of Lagrange multipliers.

Now we come to the crucial observation made by Brown and Kuchař in [18] that the system S EH + S D is indeed (partially) deparametrisable. To see this, note first, that the total Hamiltonian constraint (2.4) can be written in equivalent form

c tot = c + P 1 + q ab c dust a c dust b P 2 c + P 1 + q ab c a c b P 2 . (2.11)
On the constraint hypersurface the constraints c tot and c tot a can be solved for the momenta P and P j respectively and thus be written down in completely equivalent6 form as

ctot = P + h h = c 2 -q ab c a c b (2.12) ctot j = P j + h j h j = S a j [c a -hT ,a ] , (2.13)
where we assumed that S j ,a is non degenerate and defined its inverse S a j . As shown in [19] this condition is gauge invariant under the gauge transformations generated by (2.12) and (2.13).

At least the total Hamiltonian constraint is in deparametrised form now, because h depends only on the gravitational variables q ab , p ab . Moreover, since the constraints are linear in the dust momenta, (2.12) and (2.13) form a strongly Abelian first class constraint algebra. As a consequence also {h(x), h(y)} = 0 because h commutes with P . However, only the total Hamiltonian constraint c tot is of deparametrised form but not the total diffeomorphism constraint, so Poisson brackets between either h(x) and h j (y) or h j (x) and h j (y) will not vanish in general. So we achieved a partially deparametrised form for the coupled system of gravity and dust. Note that this can also be obtained when additional, for instance standard model, matter would be coupled to gravity and dust. The only difference in this case will be that the constraints c and c a above will then not only include gravitational contributions but consist of a sum of gravity plus the additional matter contributions. An example where additional to gravity and dust a K.G. -scalar field was considered can be found in [19,20].

We are now in the position to define distinguished coordinates on the reduced phase space defined by the first class system defined by (2.12) and (2.13). It is clear that T, S j are pure gauge and that P, P j can be solved for the gravitational (and standard matter) field variables. Therefore it is natural to introduce a four parameter family of gauge fixing conditions defined by T (x, t) = τ, S j (x, t) = σ j where τ ∈ R and σ takes values in the dust space S = S(X ) which by assumption is diffeomorphic to X . The dust space labels the geodesics and τ is an affine parameter along the geodesics.

The relational framework can now be applied and the results can be described as follows (see [19] for all the details and the derivations): Let S a j (x) be the inverse of S j ,a (x) and J(x) := det(∂S/∂x). Consider

qij (σ) = χ d 3 x|J(x)|δ(σ j -S j (x))S a i S b j q ab pij (σ) = χ d 3 x|J(x)|δ(σ j -S j (x)) S i ,a S j ,b p ab J (x) (2.14) It maybe checked explicitly that {p ij (σ), qkl (σ )} = κδ i (k δ j l) δ(σ, σ ) . (2.15)
The interpretation of (2.14) is obvious: These are the coordinate transformations of (q ab , P ab ) respectively into the dynamical coordinate system defined by x σ = S -1 (σ). One can arrive at these expressions independently by symplectic reduction which explains why (q ij , pij ) continue to be a conjugate pair.

Next let

Q jk (τ, σ) := ∞ n=0 1 n! { hτ , qjk } (n) P jk (τ, σ) := ∞ n=0 1 n! { hτ , pjk } (n) (2.16)
where we introduced hτ := S d 3 σ τ -T (σ) h(σ) with h = h(q jk , pjk ) (2.17)

It maybe checked explicitly that (2.16) has vanishing Poisson brackets with all constraints. These expressions can no longer be described in a compact form, they are hopelessly complicated to evaluate as functions of q, p. However, as stressed in the introduction, all we need is their Poisson algebra and their time evolution. To that end, set

Q ij (σ) := Q ij (τ = 0, σ), P ij (σ) := P ij (τ = 0, σ). Then it maybe checked explicitly that {P ij (σ), Q kl (σ )} = κδ i (k δ j l) δ(σ, σ ) . (2.18)
still form a canonical pair, all other Poisson brackets vanishing. Furthermore, let us define

H(σ) = C 2 -Q ij C i C j (σ) , (2.19)
where

C := c(q jk = Q jk , pjk = P jk ), C i := ci (q jk = Q jk , pjk = P jk ), (2.20)
and c, ci are just c, c a expressed in the dust coordinate system. Then it may be checked that the physical Hamiltonian generating time evolution for all observables f (Q ij , P ij ) is given by

H phys := S d 3 σ H(σ) .
(2.21)

When looking at its variation

δH phys = S d 3 σ C H δC - Q ij C j H δC i + 1 2H C i C j Q ik Q jl δQ ij (2.22) =: S d 3 σ N δC + N i δC i + 1 2 HN i N j δQ ij , (2.23)
one sees that the physical equations of motion generated by H phys are almost equivalent to the ones generated by the canonical Hamiltonian H can with the identification q ab (x) → Q ij (σ), p ab (x) → P ij (σ) modulo the following important differences: First, lapse N and shift N i are not phase space independent functions as for H can where they only encode the arbitrariness of the foliation. Rather they are observable phase space functions composed out of the elementary fields Q ij , P ij as

N := C H , N i = - Q ij C j H , (2.24)
Second, there is one additional contribution proportional to the Hamiltonian density H(σ). But H(σ) is a conserved quantity in the theory and can be freely chosen on the initial value hypersurface, so we may tune this term to alter the equations of motion as little as we like.

The physical Hamiltonian has an infinite number of conserved charges, namely energy and momentum density H(σ), C j (σ). The latter ones generate active diffeomorphisms of the dust space S, they are to be considered as symmetries of the system rather than gauge transformations generated by the passive diffeomorphisms of X . Likewise, the former are related in an intricate way to time reparametrisation invariance in General Relativity.

Spherical symmetry

Now we want to specialise the general theory to spherically symmetric spacetimes. We will work directly at the gauge invariant level and assume that what one usually measures in physical experiments are not the gauge variant three metrics q ab and their canonically conjugate momenta p ab in some unphysical coordinate system x a but rather the physical metrics Q ij and their physical canonically conjugate momenta P ij measured with respect to the physical reference frame S given by the dust fields. There is nothing to debate about the fact that whenever we perform experiments we measure physical gauge invariant quantities and not the kinematical quantities q ab , p ab . Hence we will require spherical symmetry with respect to the physical coordinate system σ.

Thus, spherically symmetric spacetimes M = R × S will be characterised by a triplet of Killing vector fields { ξ 1 , ξ 2 , ξ 3 } on S whose commutator algebra is isomorphic to the Lie algebra so(3). As usual, for such spacetimes it is always possible to find a coordinate chart in which the physical four metric G µν takes the special form

G µν = -N 2 + Q ij N i N j N i N j Q ij , (3.1)
where i, j = 1, 2, 3 and

Q ij (σ r ) = diag Λ 2 (σ r ), R 2 (σ r ), R 2 (σ r ) sin 2 σ θ (3.2) P ij (σ r ) = sin σ θ diag P Λ (σ r ) 2Λ(σ r ) , P R (σ r ) 4R(σ r ) , P R (σ r ) 4R(σ r ) sin -2 σ θ , , (3.3)
where (σ r , σ θ , σ φ ) are spherical coordinates of a physical coordinate system on the spatial slices and (Λ, P Λ ) and (R, P R ) respectively are conjugate pairs. Hence, the only non vanishing Poisson brackets are given by

{P Λ (σ r ), Λ(σ r )} = κ 4π δ(σ r , σ r ), {P R (σ r ), R(σ r )} = κ 4π δ(σ r , σ r ) . (3.4)
In contrast to the gauge variant formalism lapse N and shift N i are not arbitrary phase space independent functions but are given by (2.24). Using these fields the geometry parts of the Hamiltonian and spatial diffeomorphism constraints reduce to:

C = 1 κ sin σ θ ΛR 2 P Λ 2 8R 4 - P Λ P R 4ΛR 3 + 2 Λ 2 R R 2 + 4 Λ 2 R R - 4R Λ Λ 3 R - 2 R 2 (3.5) C σr = 1 κ sin σ θ -ΛP Λ + R P R (3.6) C σ θ = C geo σ φ = 0 , (3.7)
Hence, the physical Hamiltonian H phys , which generates the evolution of observables, specialised to the case of spherically symmetric spacetimes reads as

H phys = S d 3 σ C 2 - 1 Λ 2 C 2 σr . (3.8)

Equations of motion

Now we want to discuss the physical equations of motion for the spherically symmetric case: Physical time evolution is generated by the physical Hamiltonian (2.21), so as a first step we need to compute its variation. This was already demonstrated for the general case at the end of section 2 and now we want to specialise this result to the spherically symmetric case: For this purpose we must use the physical Hamiltonian H phys that has been specialised to the spherically symmetric case in (3.8) and compute its variation while considering Λ, R, P Λ and P R as the basic variables. This results in the following first order equations of motion:

Λ = H phys , Λ Ṙ = H phys , R ṖΛ = H phys , P Λ ṖR = H phys , P R . (4.1)
The explicit derivation of physical equations of motion from the variation δH phys of the physical Hamiltonian has been done for full GR in [19] and it was shown that suitable boundary conditions can be chosen so that boundary terms can be neglected in that calculation. This carries over to the spherically symmetric case.

A discussion of boundary conditions in the latter case is given in appendix A. Denoting derivatives with respect to τ and σ r with dot and slash respectively the first order equations of motion explicitly read as

Λ = N ΛP Λ 4R 2 - N P R 4R + (N σr ) Λ + N σr Λ (4.2) Ṙ = - N P Λ 4R + N σr R (4.3) ṖΛ = - N C(σ r ) 4πΛ - N P Λ P R 4ΛR - 4N RR Λ 2 + 4N R R Λ 2 - 4N RR Λ Λ 3 +N σr P Λ -(N σr ) 2 ΛH (4.4) ṖR = - 2N C(σ r ) 4πR + N ΛP Λ 2 2R 3 - 3N P Λ P R 4R 2 + 4N R 2 ΛR + 4N R Λ - 4N R Λ Λ 2 - 4N Λ R - 4N R Λ - 4N R Λ + 4N RΛ Λ 2 + (N σr ) P R + N σr P R . (4.5)
We obtain the same equations when we specialise the first order Hamiltonian equations for full GR given in [19] to spherical symmetry and additionally set the K.G. -scalar field contributions to zero. Formally these equations of motion coincide with the ones generated by H can for the gauge variant 3-metrics and momenta up to the term proportional to the Hamiltonian density H in (4.4), but there is one important difference: N and N σr are no free functions anymore. Rather they are phase space dependent functions given by (2.24) and choosing different N, N i is equivalent to working in a different physically distinguishable coordinate system. This again is due to the fact that Diff(S) is the group of active diffeomorphisms on S as opposed to the passive diffeomorphism on X .

The Hamiltonian density H(σ) as well as C σr are constants of motion [19]. The latter corresponds to the momentum density of the dust denoted by σr (σ), that is C σr (σ) = -σr (σ). In the following we will restrict our discussion to the case of vanishing dust momentum density. Then the shift vector N σr being proportional to C σr is vanishing. Furthermore the Hamiltonian density is given by

H = C 2 -Q ij C i C j = C
H(σ) = sin σ θ (σ r ) for (σ r ) > 0 . (4.6) From H(σ) = -P (σ) = ρ dust det(Q) we obtain ρ dust (τ, σ r ) = - sin σ θ (σ r ) √ det Q = - (σ r ) Λ(τ, σ r )R 2 (τ, σ r ) . (4.7)
Hence, we can write

C(σ r ) := dσ θ dσ φ C(σ) = -dσ θ dσ φ sin σ θ Λ(σ r )R 2 (σ r )ρ D (τ, σ r ) = -4πρ D (σ r )Λ(σ r )R 2 (σ r ) . (4.8)
For further analysis we want to transform the first order system (4.2) -(4.5) into a second order system. To this end one can solve (4.2) and (4.3) for the momenta and then use these expressions and their τ -derivatives in (4.4) and (4.5). Setting N = 1 and N σr = 0 and using (4.8) this results in

Λ Λ + R R - Λ Ṙ ΛR - Λ R Λ 3 R - Ṙ R 2 + 1 Λ 2 R R 2 + 2R Λ 2 R - 1 R 2 = - ρ D 2 , (4.9) 2 R R + 2R Λ 2 R - 2 Λ Ṙ ΛR - 2Λ R Λ 3 R = - ρ D 2 . (4.10) Furthermore we can use that H(σ) = -ρ dust det(Q) yielding - Ṙ R 2 + 1 Λ 2 R R 2 - 2 Ṙ Λ ΛR - 2R Λ Λ 3 R + 2R Λ 2 R - 1 R 2 = - ρ D 2 , (4.11)
and the vanishing of the momentum density

Ṙ - Λ Λ R = 0 .
(4.12) (4.9) -(4.11) are actually not entirely independent of each other, in fact it is sufficient to keep (4.10) and (4.11): One can check that solutions to these two equations also satisfy (4.9).

Solving the equations of motion

In this section we will discuss the solution of the equations of motion. We will mainly follow [25] for the derivation of this solution using a slightly different notation here. In order to solve (4.9) -(4.12) it is first important to notice that the (physical τ -) time dependencies of Λ(τ, σ r ) and R(τ, σ r ) are not independent from each other. We can solve (4.12) and get:

Λ(τ, σ r ) = R (τ, σ r ) 1 1 + E(σ r ) , (5.1)
where E(σ r ) is a so far arbitrary, time-independent function7 of σ r . Exploiting this, we can rewrite (4.10) and (4.11) as

2 R R + Ṙ R 2 - E R 2 = 0 (5.2) R R - Ṙ Ṙ R R + E 2R R = - ρ D 4 . (5.3)
First, by multiplying (5.2) by R 2 Ṙ we get

2R Ṙ R + Ṙ3 -E Ṙ = R(2 Ṙ R) + Ṙ( Ṙ2 -E) = d dτ R( Ṙ2 -E) = 0 ⇒ U = 0 , (5.4)
where we defined U = R( Ṙ2 -E). U does not depend on τ so we can write U (τ, σ r ) = F (σ r ) for an so far arbitrary σ r -dependent function F (σ r ). By differentiating U with respect to σ r we get

U = F = R ( Ṙ2 -E) + 2R Ṙ Ṙ -E R .
(5.5)

Dividing this by R 2 R we see that F is equal to the lefthand side of (5.2) minus 2 times the lefthand side of (5.3). Thus, we can rewrite F as

F = 1 2 R R 2 ρ D = 1 2 √ 1 + E ΛR 2 ρ D = - 1 2 √ 1 + E (σ r ) , (5.6)
where we used (5.1) in the first equality and (4.7) in the second one. Integrating the equation above once, we end up with

F (σ r ) = - 1 2 σr 0 dλ (λ) 1 + E(λ) + α =: - 1 2 M (σ r ) + 1 2 M (0) , (5.7)
where M (0) is an so far arbitrary (constant) mass. M (σ r ) can be interpreted as the effective gravitating mass (multiplied by 1/4 since the relation between the Schwarzschild radius and the central mass involves a factor of 2) of the dust inside a sphere with radial label σ r . So finally the equation

U = F yields Ṙ = ± F R + E .
(5.8)

We arrive at the following general form of the metric

ds 2 = -dτ 2 + R 2 (τ, σ r ) 1 + E(σ r ) dσ r 2 + R 2 (τ, σ r )dΩ 2 Ṙ(τ, σ r ) = ± E(σ r ) + F (σ r ) R(τ, σ r ) ρ D (τ, σ r ) = 2F (σ r ) R (τ, σ r )R 2 (τ, σ r ) , (5.9)
with E(σ r ) > -1 and F (σ r ) must be chosen such that the expression under the square root in the second line is non-negative for all R(τ, σ r ), see section 7 for further elaborations on this topic. The solution for R(τ, σ r ) for general E(σ r ) can only be given in parametric form, distinguishing the cases E > 0, E = 0, E < 0 and using the conventions of [26] one finds:

• E(σ r ) > 0 : R(τ, σ r ) = F (σ r ) 2E(σ r ) cosh(η) -1 sinh(η) -η = 2 E(σ r ) 3 2 (β(σ r ) -τ ) F (σ r ) (5.10) • E(σ r ) < 0 : R(τ, σ r ) = F (σ r ) 2(-E(σ r )) 1 -cos(η) η -sin(η) = 2 -E(σ r ) 3 2 (β(σ r ) -τ ) F (σ r ) (5.11) • E(σ r ) = 0 : R(τ, σ r ) = 3 2 √ F β(σ r ) -τ 2/3
(5.12)

Here β(σ r ), E(σ r ) and F (σ r ) are so far arbitrary functions of σ r which allow a coordinate choice by a particular form of β(σ r ) and two further physical quantities. As mentioned above F in (5.7) can be understood as the effective gravitating mass within radius σ r and E(σ r ) determines the time evolution of R as well as the local geometry. For a more detailed discussion see for instance [27]. The time τ at which R(τ, σ r ) is equal to zero is β(σ r ) and one calls τ ≥ β(σ r ) the big bang time whereas τ ≤ β(σ r ) is referred to as the recollapse time. Often one refers to these three cases as elliptic, parabolic and hyperbolic for E(σ r ) < 0, E(σ r ) = 0 and E(σ r ) > 0 respectively. The solution in (5.9) can easily be identified as the whole class of Lemaître-Tolman-Bondi (LTB) metrics (see appendix B.3), so we managed to map the problem of solving the physical τ -evolution of gauge-invariant three-metrics Q ij to the problem of analysing the LTB class of solutions in standard GR. However, it is important to note that opposed to the standard LTB models, the mass term M (σ r ) enters with the opposite sign in F (σ r ) = 1 2 (M (0) -M (σ r )). That means the effective gravitating mass will decrease when going further away from the central mass. This happens because the dust clocks, which have been chosen as a physical reference system, have negative energy.

We can fix the coordinate choice included in β(σ r ) and the mass M (0) in (5.7) by requiring that in the vacuum case the general solution above should reduce to the Lemaître solution [28] given by

ds 2 = -dτ 2 + R s R Lem (τ, σ r )) dσ r 2 + R 2 Lem (τ, σ r )dΩ 2 with R Lem (τ, σ r ) = 3 2 R s (σ r -τ ) 2/3
(5.13)

where R s = 2M G/c 2 denotes the Schwarzschild radius which simplifies in units

h = c = G = 1 to R s = 2M .
The Lemaître solution is diffeomorphic to the Schwarzschild solution and can be obtained from the latter through a coordinate transformation into a comoving coordinate system. So this metric describes the local coordinate system of a free falling observer in a spherically symmetric gravitational field originating from a central mass M, more details can be found in appendix B.1. Matching the Lemaitre with the LTB solution we obviously need to consider the case E(σ r ) = 0. This is often referred to as the marginally bound case. Furthermore, in the vacuum case, (σ r ) is zero and as a consequence F vanishes. Hence, in this case F is just equal to the constant 1 2 M (0) introduced in (5.7). Using this we obtain R(σ r , 0) = 3/2 1 2 M (0)(β(σ r ))

2/3 whereas R Lem (σ r , 0) = 3/2 √ R s (σ r ) 2/3
. Thus, we can match these two radii by simply choosing β(σ r ) = σ r and identifying M (0) = 4M because R s = 2M = 1/2M (0).

Newtonian limit

Let us have a look at the Newtonian limit of this spacetime: In [25] it was argued that R(τ, σ r ) is the quantity that should actually be identified with "Newtonian distance", for instance in the context of luminosity distance. Having this in mind let us look at equation (5.8) again. By differentiating this one with respect to τ we obtain

R = - F 2R 2 , (5.14)
which coincides formally with Newton's equation of motion for a point particle under the influence of a central mass. The crucial difference is that the effective gravitating mass F/2 is less than the central mass M = R s /2. So the gravitational field is weaker than one would expect without having the dust around.

There is one remark necessary concerning the falloff behaviour of R(τ, σ r ): One might wonder whether the boundary conditions discussed in appendix A are not violated by the explicit form of R as in (5.12). The solution to this puzzle lies in the fact that in the derivation of the falloff conditions (A.3) we assumed a coordinate system which approaches a flat spherical Minkowskian one in the asymptotic limit. In contrast, the coordinate system we are using here is the coordinate system of a freely falling observer so one cannot expect (5.12) to hold automatically. Nevertheless, we assumed that (σ r ) → 0 for σ r → ∞, so in the limit of vanishing dust density and in the marginally bound case (that is E=0) the LTB coordinates merge into Lemaitre coordinates which themselves can be transformed to Schwarzschild coordinates having the correct Newtonian limit (see appendix B.1) .

We have seen above that the exact Schwarzschild solution is obtained for F =const. and E = 0. However, it is possible to obtain the Schwarzschild solution also for certain E = 0 and F =const. The idea is to transform the LTB coordinates first into generalised Painleve -Gullstrand coordinates (GPG), introduced in [29] and then to Schwarzschild coordinates. For later comparison, let us first transform the Lemaitre solution from Lemaitre coordinates (τ, σ r ) to the ordinary Painleve -Gullstrand (PG) coordinates (τ, R) corresponding to observers moving on radial timelike geodesics labelled by Schwarzschild radial coordinate R. This yields a non -diagonal metric whose line element is of the form [30, 31]

ds 2 = -dτ 2 + dR + R s R dτ 2 + R 2 dΩ 2 (5.15)
where we used τ for the Lemaitre and PG time coordinate. We present in appendix B.2 how to transform from Schwarzschild to PG coordinates. For a pedagogical introduction to PG coordinates and more details see for instance [32]. Often one uses Kruskal coordinates to obtain a maximally extended Schwarzschild spacetime. The PG coordinates do not cover the total Kruskal manifold. In the form given in equation (5.15) they cover the black hole horizon and the black hole singularity at R = 0 hence the physically interesting part of the manifold for our discussion. If we choose the sign of Ṙ = ± E + F/R to be negative then we obtain another set of coordinates with a line element similar to that in (5.15) but with a minus sign in front of the off diagonal term. These PG -coordinates will then describe the so called "white hole region" of the Kruskal manifold. One of the properties of Kruskal coordinates is that R is given only implicitly therefore for our purpose of discussing the Newtonian limit GPG coordinates are more appropriate. One of the interesting properties of the ordinary PG metric shown in (5.15) is that the spatial hypersurfaces (that is dτ = 0) are flat since then ds 2 = dR 2 + R 2 dΩ 2 and all information about the curvature is encoded in the shift vector given by N = (N R := R s /R, 0, 0). Furthermore, asymptotically (R → ∞) the PG metric coincides with the Minkowski metric. Now we transform the LTB coordinates (τ, σ r , θ, φ) into GPG coordinates (τ, R, θ, φ). Following [29] this yields the metric:

ds 2 = -dτ 2 + dR + E(τ, R) + F (τ,R) R dτ 2 1 + E(τ, R) + R 2 dΩ 2 (5.16) subject to E(τ, R) > -1.
The functions E and F that are functions of σ r only in LTB coordinates now become functions depending on τ and R in GPG coordinates. Therefore the time dependence of E and F in these coordinates is restricted by the partial differential equations

∂E(τ, R) ∂τ -E(τ, R) + F (τ, R) R ∂E(τ, R) ∂R = 0 ∂F (τ, R) ∂τ -E(τ, R) + F (τ, R) R ∂F (τ, R) ∂R = 0
(5.17)

More details about the transformation from LTB to GPG coordinates can be found in appendix B.4. We will not comment on the full class of solutions of this system of PDE's but just remark that e.g. (local) analytic solutions can be found by providing analytic initial data F (0, R), E(0, R) and determining the coefficients of the Taylor expansion (in terms of τ ) of E(τ, R), F (τ, R) by taking higher derivatives of (5.17) at τ = 0 (Kovalevskaja method). For instance, a trivial solution is given by taking E =const. and F =const.

In the special case E(τ, R) = 0 and F (τ, R) = R s = const the GPG metric coincides with the PG metric justifying the name generalised Painleve -Gullstrand metric. In contrast to the PG solution in general the spatial hypersurfaces of the GPG metric are no longer flat due to the 1/(1 + E) factor in front of dR 2 . So one could think that in the limit of vanishing dust density (t, R) → 0 a transformation to Schwarzschild coordinates is only possible in the marginally bound case E = 0. As discussed in [29] this is not the case and we summarise their discussion in the following: Let us consider the case of vanishing dust density (t, R) = 0. Since (τ, R) ∼ F it follows F = R s . The equation for F in (5.17) is thus trivially satisfied. For each function E satisfying the equation in (5.17) there exists a transformation from GPG coordinates (τ, R, θ, φ) to Schwarzschild coordinates (T, R, θ, φ) given by

∂τ ∂T 2 = 1 + E ∂τ ∂R = E + Rs R 1 -Rs R (5.18)
Thus one obtains a family of coordinate transformations parametrised by the functions E. Indeed, the integrability condition for the system (5.18) leads back to the first condition in (5.17) (notice that one has to write E(T, R) ≡ E(τ (T, R), R) in order to derive it). Therefore by first transforming the LTB to GPG coordinates it is possible to transform to Schwarzschild coordinates in the limit of vanishing dust density for all (allowed) functions E. Thus we can transform to Schwarschild coordinates not only in the marginally bound case E = 0 but also in the elliptic (E < 0) and hyperbolic (E > 0) cases provided F =const. Consequently, the Newtonian limit is correctly implemented for all values of E and F =const.

Semistatic properties

Now we want to analyse whether the physical spacetime (M, G) under consideration is semistatic in a certain sense or not. We already know that the physical metric G µν takes the form of a Lemaître metric in the limit (σ r ) → 0 which in turn is just a Schwarzschild spacetime written in comoving coordinates (see appendix B.1). So one would expect to recover its static properties in some limit.

Let us have a look at the Killing equation

(L ξ G µν ) ! = 0 (6.1)
for the observable metric G µν and examine whether there exists a timelike Killing vector field ξ. For the Lemaître metric such a Killing vector field is given by ξ = [1, 1, 0, 0] T , at least outside the event horizon, so we would expect to find something similar here. We will start with the ansatz

ξ = ξ τ (τ, σ r ), ξ σr (τ, σ r ), 0, 0 T . (6.2)
For the physical metric described above (6.1) reduces to ξτ = 0 (6.3)

ξ τ Ṙ + ξ σr R + R (ξ σr ) = 0 (6.4) ξ τ Ṙ + ξ σr R = 0 (6.5) -(ξ τ ) + R 2 ξσr = 0 . (6.6)
From ( 6.5) we get (6.7) and using (6.3) we obtain

ξ σr = - Ṙ R ξ τ ,
ξσr = - R R + Ṙ Ṙ R 2 ξ τ .
(6.8) (6.6) can be solved for (6.9) and using this in (6.7) we see that

(ξ τ ) = -R R + Ṙ Ṙ ξ τ ,
(ξ σr ) = - Ṙ R + ṘR R 2 + Ṙ R - Ṙ2 Ṙ R ξ τ . (6.10)
Putting all this together we obtain a condition on the solution R R -Ṙ Ṙ ! = 0 , (6.11) and using the explicit form of the solution (5.12) we see that this holds only if (6.12) This means that there exists a timelike Killing vector field ξ only if (σ r ) = 0, that is if the dust matter fields are vanishing. Nevertheless, as one can easily check, the metric is invariant under the action of the vector field

F (σ r ) ! = 0 .
ξ 0 := 1, - Ṙ R , 0, 0 T (6.13)
up to terms of at least O( (σ r )). So with

(L ξ 0 G) µν = O( (σ r )) (6.14)
there is a precise sense in which the physical metric G µν can be called "semistatic". For vanishing (σ r ) this would be an exact symmetry and the metric would be isomorphic to the standard Schwarzschild metric. (σ r ) = 0 would be inconsistent because we are real dust fields as dynamical clocks rather than ideal ones and vanishing (σ r ) would mean that the clocks do not carry any energy at all8 . Nevertheless, (σ r ) is a free function in the theory and we can choose it arbitrarily small9 . So by tuning (σ r ) in the right way we can get as close to a static spacetime as we wish.

Discussion of Singularities

In this section we will discuss the possible singularities which might appear for the metric given in (5.9) and displayed again below:

ds 2 = -dτ 2 + R 2 1 + E(σ r ) dσ r 2 + R 2 dΩ 2 . (7.1)
As is well known from earlier studies concerning LTB spacetimes (see for example [33,27]) there appear two different kinds of singularities: Recalling that the dust density was given by ρ dust = F R R 2 there are potentially two cases when it can diverge. The first one, if R(τ, σ r ) = 0, is called a collapse singularity in the literature, and the second, if R (τ, σ r ) = 0, is known as a shell crossing singularity. It was pointed out by [34] for the first time that in the latter case an appropriate decay behaviour for F when σ r → ∞ can avoid the divergence of ρ dust in the shell crossing case. From the line element above we can read off that in the case of a collapse singularity the metric components Q θθ and Q φφ vanish while for a shell crossing singularity only Q σrσr is zero.

Shell crossing singularities

For standard LTB spacetimes with positive dust densities one can always choose the arbitrary functions E(σ r ), F (σ r ) in such a way that shell crossing singularities will not appear [33]. This holds not only for the marginally bound case (E(σ r ) = 0) but also for the hyperbolic (E(σ r ) > 0) and elliptic (-1 < E(σ r ) < 0) case. In our framework the dust energy density enters with negative sign. Therefore F (σ) has negative sign. Furthermore we have the requirement F/R + E > 0. Consequently we arrive at slightly different conditions on E(σ r ), F (σ r ) in order to avoid shell crossing singularities. The analysis for E(σ r ) = 0 involves the parametric solutions for R(τ, σ r ) shown in (5.10) and (5.11). A singularity discussion becomes more involved in these two cases and we will restrict ourselves to the marginally bound case (E(σ r ) = 0) here, similar results can also be obtained for the hyperbolic and elliptic case.

If E = 0 then Q σrσr is given by R 2 (τ, σ r ) = 1 4R(τ, σ r )F (σ r ) F (σ r )(σ r -τ ) + 2F (σ r ) 2 . (7.2)
Considering the constraints F > 0, F < 0, σ r -τ ≥ 0, in order to avoid shell crossing, i.e. R = 0 for all τ, σ r we obviously must have

0 < -F < 2F σ r -τ or -F > 2F σ r -τ > 0 (7.3)
for all σ r , τ . Since F only depends on σ r , for any given σ r we can choose σ r -τ arbitrarily small so that for any choice of F the second possibility can be violated. Hence the only dynamically stable condition is

0 < -F < 2F σ r -τ (7.4) Since σ r ≥ σ r -τ ≥ 0, this condition is certainly implied by 0 < -F < 2F σ r ⇔ 0 < [ln(F σ r 2 )] (7.5)
Since the logarithm is an increasing function, we obtain that F σ r 2 should be an increasing function while F should be a decreasing function. This leaves us with a large class of possible F , for instance F (σ r ) = Rs 1+σr which also fulfills F ≤ R s . Inserting this form of F into R and R yields the following density for the dust

ρ dust = - 2 3 1 1 + 1 2 (σ r + τ ) (σ r -τ ) (7.6)
The dust density diverges at τ = σ r as expected because this is exactly the singularity at R = 0 that can not be avoided, see also discussion in the next section. If we take the limit σ r → ∞ for fixed values of τ then ρ dust → 0. Hence, ρ dust shows a physically reasonable behaviour.

Collapse singularities

The second kind of singularities are those where R = 0 and therefore besides the dust density ρ dust also the expression for Ṙ2 in (5.9) becomes singular. These singularities occur for all three cases -marginally bound, hyperbolic and elliptic -when τ → σ r . Having collapsing dust shells in mind this is exactly the moment when the dust shell labelled by σ r reaches the singularity. Note that close to R = 0 the expression for Ṙ2 is dominated by the E = 0 regime and thus we will again restrict our discussion to the marginally bound case. Looking at the explicit form of R(τ, σ r ) written down in (5.12) we can analyse how the different sign of the dust alters the behaviour during the collapse. In contrast to the shell crossing singularity there is no way to get around this collapse singularity for the system gravity plus pressureless dust. As long as we do not introduce additional matter fields and take their non-gravitational interactions into account10 we end up with a singularity after τ = σ r , just as in the gauge variant framework using standard LTB -models.

Finally we want to summarise all the physical selection criteria for (σ r ) in the marginally bound case we have encountered so far: First, due to the special role of the dust fields in our formalism, we demand F < 0 or equivalently > 0. Second we need to ensure that 0 < F (σ r ) ≤ R s , otherwise (5.12) is not a solution anymore; in terms of this means that ∞ 0 dσ r (σ r ) < 2R s . Third, we need to choose (σ r ) such that

|F (σ r )| = -F (σ r ) < 2F (σr)
σr is fulfilled in order to not run into any shell crossing singularities. Given these conditions the physical metric G µν is well defined until τ → σ r when a free falling observer approaches the central singularity.

Conclusions

The task of finding Dirac observables for General Relativity involves the construction of quantities that commute with all constraints of General Relativity. Dirac observables are thus the associated to gauge invariant objects for General Relativity where the gauge group is closely related to Diff(M ), the group of diffeomorphism of the underlying manifold M . The reason why it is more complicated to construct gauge invariant quantities in the framework of General Relativity is that the mathematical structure of Diff(M ) is richer than the structures of the gauge groups used for instance in the standard model of particle physics. In the context of the Relational formalism one can at least formally construct Dirac observables, with respect to chosen clocks, one for each occurring constraint. Choosing these clocks means choosing an observer that is dynamically coupled to the system. Considering the case of vacuum gravity these clocks must necessarily be four components of the four metric g µν . If one applies the techniques developed in the relational framework and tries to compute (Dirac) observables for pure GR then one realises that the physical time evolution of those observables does not resemble the Einstein equations. This is because their dynamics is described by an observer who sits in a laboratory whose motion through spacetime is defined through the dynamics of these four metric components. Of course, theoretically there always exist a coordinate transformation from this observer to for instance a free falling observer, however, practically such a transformation will be hard to find. Fortunately, for certain types of matter coupled to gravity, the constraints, consisting of the gravitational and the matter contribution, can be rewritten in (partially) deparametrised form. Hence, these enlarged systems fall into the class of deparametrisable theories for which the question of observables can be addressed technically easier then in the general case. This idea was used in [19,20] where instead of considering pure gravity (i.e. the Einstein-Hilbert action) one considers gravity plus pressureless dustmatter fields (i.e. Einstein-Hilbert plus dust action). The dust fields become the clocks of the system and correspond to a dynamically coupled free falling observer. One obtains physical equations of motion for the observable 3-metric Q jk and observable momenta P jk in the dynamical reference frame defined through the dust fields. Of course, when introducing a dynamically coupled observer, we have to ensure that the occurring fingerprints of the observer are still in agreement with experimental data.

That this is the case for the cosmological sector was already shown in [19,20] and in this article we demonstrated that this framework also describes gravitational physics in the spherically symmetric sector to arbitrary precision. We showed that within this sector the dynamical evolution of Q ij , P ij , generated by a physical Hamiltonian H phys , is in one-to-one correspondence with the class of Lemaitre-Tolman-Bondi (LTB) solutions for standard GR. Interpreted in our language, the choice of lapse and shift considered for the LTB solutions correponds to a gauge fixing of the (spherically symmetric) spacetime diffeomorphism invariance whose gauge invariant extension is precisely induced by our choice of clocks. In addition, while in the usual LTB framework one starts with a spherically symmetric, pressure free, perfect fluid Ansatz for the energy momentum tensor whose dynamics is then derived from the Bianchi identity, here we start with a fully covariant matter Lagrangian whose equations of motion then reduce to the usual ones. In other words, our framework provides a Lagrangian underpinning of the usual LTB framework. That this works so well is non trivial as we discuss in appendix C.

The LTB class of solutions has been carefully investigated in the literature and its dynamics in the comoving frame is well understood. One might object that that the LTB class does not contain static solutions, at least for non-vanishing dust energy which is mandatory if one wants to use the dust fields as clocks. But this had to be expected when taking into account the influence of a realistic observer (i.e. an observer which is dynamically coupled to the system as opposed to a mere test observer). In a sense one has to abandon the idealisation of a static (vacuum11 ) spacetime when describing physics in terms of observable quantities.

Nevertheless, we showed that there exists a well defined notion of a semistatic spacetime and one can get as close to the standard Schwarzschild solution as desired by appropriately choosing certain constants of motion (which are related to the dust energy density). Additionally, we discussed that by transforming the LTB system to generalised Painleve -Gullstrand coordinates and considering the limit of vanishing dust density there exists a well defined coordinate transformation to Schwarzschild coordinates for the elliptic, hyperbolic and marginally bound case. Consequently, in the limit where the dust density can be neglected the Newtonian limit is correctly implemented. Finally, although we consider phantom dust rather than usual dust, there exists a range of solutions for which shell crossing singularities are avoided while the collapse singularity is unavoidable.

To conclude, the framework presented in [19,20] seems to be compatible with observations, at least in the cosmological and spherically symmetric sectors which are the most relevant analytically solvable ones when it comes to phenomenological applications. This framework might also be useful when it comes to quantising General Relativity. The fact that the constraints have already been solved at the classical level opens the door for a reduced phase space quantisation as opposed to the Dirac programme which is usually employed in Loop Quantum Gravity [22,23]. First steps into this direction have already been performed in [21] and a detailed analysis of this framework in the context of spherical symmetry is the subject of future work [24].

A Boundary conditions

We are dealing with asymptotically flat spacetimes and hence must impose suitable boudary conditions. For the full theory, these boundary conditions for the gauge invariant observables as obtained via Brown -Kuchař dust reduction were discussed extensively in [19]. Here we need their reduction to spherical symmetry. Consider, as in the main text, a spherically symmetric coordinate system as (σ r , σ θ , σ φ ). Tensor indices on the dust manifold S are denoted by i, j, k, • • • = 1, 2, 3. The reduced observables are Q ij (σ) and their canonically conjugate momenta P ij (σ), both of which are observables in the sense described above. As usual, spherical symmetry constrains these fields to the following non vanishing components and coordinate with our dust time and radial coordinate. Schwarzschild coordinates are denoted by (T, R). The (generalised) Painleve -Gullstrand hybrid coordinates are (τ, R). The Lemaitre, Schwarzschild and strict Painleve -Gullstrand solutions are nothing else than coordinate transformations of the static vacuum Schwarzschild solution into comoving coordinates (on restricted patches of the fully extended Kruskal spacetime). The LTB family are not vacuum solutions and are expressed most easily in Lemaitre coordinates. Transforming the non trivial LTB solutions (i.e. non vanishing dust energy density) into Schwarzschild coordinates is not possible without picking up a non vanishing shift since these spacetimes are neither stationary nor static. However, there is a notion of semi staticity as elaborated on in the main text.

B.1 Lemaître solution

Starting from the usual Schwarzschild solution for vacuum spacetimes one can perform a coordinate transformation into comoving coordinates and arrive at what is known as the Lemaître solution [28].

Let us start with the Schwarzschild solution in a spherical coordinate chart (T, R, θ, φ), where T and R approach the usual Minkowskian temporal and radial coordinates of an observer located at spatial infinity. In this coordinate system the line element can be written as

ds 2 = -1 - R s R dT 2 + 1 1 -Rs R dR 2 + R 2 dΩ 2 , (B.1)
where R s = 2M G/c 2 is the Schwarzschild radius, M the central mass and dΩ 2 := dθ 2 + sin 2 θdφ 2 the area element on the unit sphere. That means the metric components are given by

g Schw µν := diag -1 - R s R , 1 1 -Rs R , R 2 , R 2 sin θ , (B.2)
Then we perform the following coordinate transformation (B.4) or in integrated form

dτ := dT + R s R 1 1 -Rs R dR (B.3) dσ r := dT + R R s 1 1 -Rs R dR ,
τ (T, R) = T + 2 R s R -R s log   1 + R Rs 1 - R Rs   (B.5) σ r (τ, R) = t(T, R) + 2 3 R R s R . (B.6)
Obviously this coordinate transformation is only valid for R = R s where the Schwarzschild coordinates are not well defined. The inverse coordinate transformation is given by

R(τ, σ r ) = 3/2 R s (σ r -τ ) 2/3 (B.7) T (τ, σ r ) = τ -2 R s R(τ, σ r ) + log 1 + R(τ, σ r ) R s -log 1 - R(τ, σ r ) R s , (B.8)
or in differential form (B.10) This leads to the following line element for a vacuum spacetime in the coordinate chart (τ, σ r , θ, φ)

dT = 1 1 -Rs R dτ + 1 1 -Rs R dσ r (B.9) dR = - R s R dτ + R s R dσ r .
ds 2 = -dτ 2 + R s R(τ, σ r ) dσ r 2 + R 2 (τ, σ r )dΩ 2 . (B.11)
This solution is known as Lemaître metric and as one can easily read of the metric components in this coordinate chart

g Lem µν = diag -1, R s R(τ, σ r ) , R 2 (τ, σ r ), R 2 (τ, σ r ) sin 2 θ . (B.12)
This metric describes the local coordinate system of an observer who is freely falling under the the influence of the central mass. It it not obvious (as for the Schwarzschild solution) right from the beginning that this solution describes a static spacetime, but one can easily calculate that for R > R s there exists a timelike, hypersurface-orthogonal Killing vector ξ K ∝ [1, 1, 0, 0] T .

B.2 Painleve -Gullstrand -Solution

In order to transform the Schwarzschild solution shown in equation (B.1) and (B.2) respectively to Painleve -Gullstrand coordinates, we introduce an observer that moves along ingoing radial, timelike geodesics of the Schwarschild spacetime. Thus the observer's time is identical with the Lemaitre time τ introduced in the last section in equation (B.3) in differential form and in equation (B.5) in integrated form. In contrast to the Lemaitre metric the Painleve -Gullstrand solution [30,31] has as the radial component the Schwarzschild R.

Hence, we want to perform a transformation from the Schwarzschild coordinates (T, R, θ, φ) to the Painleve -Gullstrand coordinates (τ, R, θ, φ). This coordinate transformation has the following differential form

dτ := dT + R s R 1 1 -Rs R dR (B.13)
The vacuum spacetime in the coordinate chart (τ, R, θ, φ) is then given by the following line element [30,31]

ds 2 = -dτ 2 + dR + R s R dτ 2 + R 2 dΩ 2 (B.14)
The components of the Painleve -Gullstrand metric are given by

g P G µν =       -1 Rs R 0 0 Rs R 1 0 0 0 0 R 2 0 0 0 0 R 2 sin 2 θ       (B.15)
This metric is no longer diagonal but still has a simple form. In particular the spatial hypersurfaces associated to this spacetime are flat because for dt = 0 we get ds 2 = dR 2 + R 2 dΩ 2 . All information about the curvature is encoded in the shift vector N = (N R = R s /R, 0, 0). As in the case of the Lemaitre solution the coordinate transformation from Schwarzschild to Painleve -Gullstrand coordinates can only be performed when R = R s . This coordinate transformation corresponds to a negative Ṙ in Schwarzschild coordinates (in general Ṙ = ± Rs R ), R decreases in time since we are considering observers moving along ingoing geodesics. Choosing the opposite sign for Ṙ yields the transformation

dτ := dT - R s R 1 1 -Rs R dR (B.16)
For this form of dt we end up with the following line element

ds 2 = -dτ 2 + dR - R s R dt 2 + R 2 dΩ 2 (B.17)
This line element corresponds to the so called "white hole region" of the extended Schwarzschild spacetime.

B.3 Lemaître-Tolman-Bondi solutions

The Lemaître-Tolman-Bondi solution (LTB) is a family of exact solutions to Einstein's field equations (see e.g. [28,36,25]) that describe dynamics of a spherically symmetric spacetime filled with inhomogeneous, pressureless dust with energy momentum tensor T µν = ρ D U µ U ν where ρ D is the dust's energy density and U µ = [1, 0, 0, 0] T its velocity vector field in comoving coordinates12 .

A general solution to this problem (in a spherical coordinate chart (τ, σ r , θ, φ)) is given by

ds 2 = -dτ 2 + R 2 (τ, σ r ) 1 + E(σ r ) dσ r 2 + R 2 (τ, σ r )dΩ 2 ρ D (τ, σ r ) = 2F (σ r ) R 2 (τ, σ r )R (τ, σ r ) Ṙ(τ, σ r ) = ± E(σ r ) + F (σ r ) R(τ, σ r ) , (B.18)
where slash and dot denote derivatives with respect to σ r and τ respectively. So one can characterise a particular model by choosing particular σ r -dependent functions E(σ r ) and F (σ r ).

Then the physical radius R(τ, σ r ) is fixed up to an arbitrary function β(σ r ) which characterises different initial conditions for R(σ r , 0). The quantity E(σ r ) determines the evolution of R as well as the local geometry and F (σ r ) is related to the mass inside a shell with radial label σ r , see [25] or for a more detailed explanation of the physical relevance of this model. For different values of E(σ r ) one obtains different solutions, which using the notation of [26] can be given in parametric form as follows:

• E(σ r ) > 0:

R(τ, σ r ) = F (σ r ) 2E(σ r ) (cosh η -1) (sinh η -η) = 2E 3/2 (σ r )(β(σ r ) -τ ) F (σ r ) (B.19) • E(σ r ) = 0: R(τ, σ r ) = 3 2 F (σ r )(β(σ r ) -τ ) 2/3 (B.20) • E(σ r ) < 0: R(τ, σ r ) = F (σ r ) 2(-E(σ r )) (1 -cos η) (η -sin η) = 2(-E(σ r )) 3/2 (β(σ r ) -τ ) F (σ r ) (B.21)
The LTB-family of solutions is widely used to describe astrophysical situations, for example it can be used to model the gravitational collapse of a (only gravitationally interacting) matter cloud. Furthermore it can be applied to cosmological models which go beyond the standard assumptions of homogeneity in ordinary FRW-evolution.

B.4 Generalised Painleve -Gullstrand Solutions

Following [29] we present in this section how the LTB metric can be transformed into a metric expressed in terms of generalised Painleve -Gullstrand coordinates (GPG). As for the ordinary Painleve -Gullstrand coordinates we want the LTB -time and the GPG -time to coincide and for the radial component we take the function R(τ, σ r ) occurring in front of dΩ 2 in the LTB line element in equation (B.18). Hence, the transformation from LTB -coordinates (τ, σ r , θ, φ) to GPG -coordinates denoted by (τ, R, θ, φ) is in differential form given by

dR = ∂R ∂τ dτ + ∂R ∂σ r dσ r = Ṙdτ + R dσ r (B.22)
Let us consider the following general ansatz for the line element

ds 2 = -Xdτ 2 + Y dR 2 + Zdτ dR + R 2 dΩ 2 (B.
X -Y Ṙ2 -Z Ṙ = 1 (B.24) Y R 2 = R 2 1 + E (B.25) ZR + 2Y ṘR = 0 (B.26)
This system of equations has the following solutions for X, Y and Z

X(τ, R) = 1 - Ṙ2 1 + E(τ, R) , Y (t, R) = 1 1 + E(τ, R) , Z(τ, R) = - 2 Ṙ 1 + E(τ, R) (B.27)
where substitution for Ṙ via (D.1) is being understood. The function E(σ r ) occurring in the LTB solution becomes a function of E(τ, R) when expressing σ r in terms of τ, R. The equation for Ṙ2 in LTB coordinates given in equation (B.18) has the following expression in the GPG coordinates

Ṙ2 = E(τ, R) + F (τ, R) R ⇔ Ṙ = ± E(τ, R) + F (τ, R) R . (B.28)
Since we want to cover the black hole region of the extended Schwarschild spacetime, we choose similar to the case of the ordinary GP coordinates a negative square root for Ṙ and obtain the following line element

ds 2 = -dτ 2 + dR + E(τ, R) + F (τ,R) R dτ 2 1 + E(τ, R) + R 2 dΩ 2 (B.29)
This line element is well defined for all values of τ, R for which E(τ, R) > -1 and E(τ, R) + F (τ, R)/R ≥ 0. The opposite sign choice for Ṙ leads to a minus sign in front of the dt -term in the bracket of the second term and describes, as before, the generalised "white hole region" of the spacetime. Now when solving the ADM equations that lead to the GPG metric one obtains an evolution equation for the shift vector N (τ, R) = (N R (τ, R), 0, 0) given by

L n (N R (τ, R)) 2 - F (τ, R) R = 0 with n µ = (1, -N R , 0, 0) (B.30)
being the unit vector normal to the spatial hypersurfaces of that spacetime. This is a second order equation for N R because as shown in [29] the term F (τ, R)/R is related to the shift vector component by 2RL n (N R )(τ, R) = F (τ, R)/R. See [29] for more details. From equation (B.29) we can easily read off the explicit form of the shift vector

N R (τ, R) = E(τ, R) + F (τ, R) R (B.31)
Consequently, the equation for the lapse carries over to an equation for E(τ, R) given by

L n (E) (τ, R) = 0 ⇔ ∂E(τ, R) ∂τ -E(τ, R) + F (τ, R) R ∂E(τ, R) ∂R = 0 (B.

C Covariant analysis of spherically symmetric gravity plus pressureless dust

For the sake of completeness, we analyse the dynamics of pressureless dust-matter coupled to gravity in a spherically symmetric setting using the usual covariant framework of GR. The metric g µν is a priori not a gauge invariant object and the "dynamics" generated by Einstein's equations has to be interpreted as gauge transformations in the strict sense. In order to make the framework physically meaningful, we must fix the spacetime diffeomorphism freedom. We therefore fix a comoving coordinate system with respect to which the lapse is unity and the shift vanishes. In the Hamiltonian language, this completely fixes the gauge freedom generated by the radial diffeomorphism and Hamiltonian constraints respectively. The comoving coordinates will be denoted by (t, r) in order to emphasise that they are measuring proper time t along ideal test observer geodesics labelled by r. In the usual LTB framework, and in contrast to the Brown -Kuchař framework, there is no "observer Lagrangian" that actually models these observers and their graviational interaction and whose proper time and geodesic label we have denoted (τ, σ r ) throughout the text. Rather, one constructs a spherically symmetric, pressure free, perfect fluid energy momentum tensor whose Lagrangian origin remains obscure and whose dynamics is simply induced by the Bianchi identity of the Einstein equations.

In what follows we will see that one ends up with an exact mathematical match between the two frameworks although their conceptual starting points are quite different, upon identifying (τ, σ r ) := (t, r). At first sight this may seem mathematically not too surprising because the energy momentum tensor of the Brown -Kuchař Lagrangian has a perfect fluid form whose pressure is constrained 13 to vanish. However, since the Brown -Kuchař Lagrangian involves altogether eight dust fields to begin with and displays a complicated gauge symmetry involving first and second class constraints, it is after all not straightforward to see that one obtains a perfect match. In particular, the velocity field of the Brown -Kuchař Lagrangian is a complicated aggregate composed out of dust fields and a priori cannot be prescribed to take a distinguished form.

We will assume spherical symmetry, therefore we can make the following ansatz for the line-element ds 2 : ds 2 = -dt 2 + Λ 2 (t, r)dr 2 + R 2 (t, r)dθ 2 + R 2 (t, r) sin 2 θdφ 2 .

(C.1)

This means the metric components are given by:

g µν = diag -1, Λ 2 (t, r), R 2 (t, r), R 2 (t, r) sin 2 θ (C.2)
Now we compute the Christoffel-symbols Γ λ µν := 1 2 g λσ g σµ,ν + g σν,µ -g µν,σ of g µν where we used the abbreviation g µν,σ := ∂ ∂x σ g µν . The Christoffel-symbols are given by:

Γ t =     0 0 0 0 0 Λ Λ 0 0 0 0 R Ṙ 0 0 0 0 R Ṙ sin 2 θ     (C.
Γ θ =      0 0 Ṙ R 0 0 0 R R 0 Ṙ R R R 0 0 0 0 0 -sin θ cos θ      (C.7) (C.8) Γ φ =      0 0 0 Ṙ R 0 0 0 R R 0 0 0 cot θ Ṙ R R R cot θ 0      . (C.9)
This leads to the following non-vanishing components of the Ricci-curvature tensor 13 That is, the vanishing of the pressure is an equation of motion and not put in by hand.

R µν = Γ ρ µν,ρ -Γ ρ µρ,ν + Γ ρ µν Γ σ ρσ -Γ σ µρ Γ ρ νσ :

R tt = - Λ Λ -2 R R R rr = Λ Λ -2 R R + 2Λ Λ Ṙ R + 2Λ R ΛR R θθ = R R - RR Λ 2 + Ṙ2 - R 2 Λ 2 + RΛ R Λ 3 + R Λ Ṙ Λ + 1 R φφ = sin 2 θR θθ R tr = -2 Ṙ R + 2 ΛR ΛR .
(C.10)

The Ricci-scalar R := g µν R µν then takes the form

R = 2 Λ Λ + 4 R R -4 R Λ 2 R + 2 Ṙ R 2 - 2 Λ 2 R R 2 + 4 Λ Ṙ ΛR + 4 Λ R Λ 3 R + 2 R 2 (C.11)
Now we want to couple matter to the system, more precisely we will use inhomogeneous pressureless dustmatter. Assuming spherical symmetry the pressure free dust stress-energy tensor is given by

T µν = ρ D (t, r)U µ U ν , (C.12)
where ρ D (t, r) can be interpreted as the dust energy density and U µ is the dust velocity vector field which in comoving coordinates takes the form U µ = δ µ t . Having collected all ingredients we can now write down Einstein's equations R µν -1/2 (4) Rg µν = κ/2T µν with κ = 16πG/c 4 for the system under consideration:

- 2R Λ 2 R - 1 Λ 2 R R 2 + Ṙ R 2 + 2 Λ R Λ 3 R + 2 Λ Ṙ ΛR + 1 R 2 = κ ρ D 2 tt-comp.
-2 One can easily see that this system of partial differential equations coincides formally with the one obtained in section 4, equations (4.9) -(4.12). So we can map the problem of finding spherically symmetric solutions in the framework of gauge invariant observables to the problem of finding spherically symmetric solutions for gravity coupled to pressureless dust-matter in the usual framework of Einstein's General Relativity in the comoving gauge N = 1, N r = 0. The system of solutions to this system of partial differential equations is the so called Lemaître-Tolman-Bondi family (see appendix B.3).

RΛ 2 R -Λ 2 Ṙ R 2 + R R 2 - Λ 2 R 2 = 0 rr-comp. -R R + RR Λ 2 - R 2 Λ Λ - RΛ R Λ 3 - R Λ Ṙ Λ =

  23) where X, Y, Z are functions of τ and R. Considering the explicit form of dR in equation (B.22) and comparing with the LTB line element in equation (B.18) we obtain the following conditions for the functions X, Y and Z

  32)Furthermore, by applying the Lie derivative L n onto (N R ) 2 and using its explicit expression in terms of E, F and R shown in equation (B.31) we obtainL n (F ) = N R 2RL n (N R )(τ, R)the GPG metric only those functions E and F are allowed that satisfy the partial differential equations in (B.32) and (B.34).

  identity (energy momentum conservation) we find∇ µ T µν = -δ t ρ D ∂ ∂t [ln(ρ D ΛR 2 )] = 0 (C.14)which obviously constrains ρ D to have the form -(r)/[Λ(t, r)R 2 (t, r)] for some free function of r only.

  and we automatically get unit lapse N = 1. This means that equations (4.2) -(4.5) simplify significantly, all terms proportional to N σr , N and N vanish. Using that H(σ) is a constant of motion, meaning that it does not evolve in physical time τ , we can write

See [2] or[3] for a detailed description how to deal with gauge symmetries in physical theories.

In the absence of boundaries such as in asymptotically flat situations.

Whenever we talk of clocks we mean devices (fields) to measure temporal and spatial distances.

A system of constraints CI is said to be deparametrisable if one can find a local coordinate chart on phase space with two mutually commuting sets of canonical pairs denoted by (q a , pa) and (T I , PI ) such that in this chart the constraints can be written in the locally equivalent form CI = PI + hI where hI depends only on (q a , pa). Equivalence has to be understood in the sense that the constraint hypersurface, spanned by the deparametrised constraints is exactly the same as the one spanned by the original set of constraints.

By equivalence we mean that the constraint hypersurface generated by both constraints are the same.

It will become clear later on why we have chosen this rather complicated form for the integration constant.

The dust part of the Hamiltonian constraint would vanish.

There are certain restrictions on (σr) which have to be fulfilled in order to avoid singularities, see section 7 for further comments on this issue.

In phenomenological matter models this is usually done by introducing matter with non vanishing pressure. This pressure is mandatory in order to describe stable configurations such as stars. In the gauge invariant formalism discussed in this article one would essentially follow the same steps, but one has to take into account the additional component ρ dust when setting up the equation of state. For non-collapsing configurations ρ dust could be tuned arbitrarily small.

These considerations do not necessarily hold for more realistic models with additional matter fields, i.e. equilibrium states for complex systems such as stars still exist, just the point of equilibrium will be slightly shifted due to the influence of the dust.

See appendix C for a covariant derivation of the equations of motion for this model.

Acknowledgements

K.G. wants to thank the Perimeter Institute for Theoretical Physics where part of this work was completed. J.T. wants to thank Bianca Dittrich for discussions concerning different choices of clock variables in the framework of complete observables. Research performed at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

† johannes(dot)tambornino(at)aei(

dependence respectively Q ij (σ r ) = diag Λ 2 (σ r ), R 2 (σ r ), R 2 (σ r ) sin 2 σ θ (A.1) P ij (σ r ) = sin σ θ diag P Λ (σ r ) 2Λ(σ r ) , P R (σ r ) 4R(σ r ) , P R (σ r ) 4R(σ r ) sin -2 σ θ , .

(A.2)

Following [35], in particular the parity conditions derived there, the following decay behaviour is sufficient to guarantee a well defined symplectic sytructure

In order to derive well defined equations of motion one should also make sure that the Hamiltonian is finite and functionally differentiable. In contrast to the gauge variant framework, it is not the canonical Hamiltonian (with independent lapse and shift fields) but rather the physical Hamiltonian (with prescribed, field dependent lapse and shift) that one has to consider. While their variations are algebraically almost identical, the field dependence of lapse and shift prevents one from adding the usual ADM counterterms under the usual decay behaviour.

In [19] it was already emphasised that, due to the dynamical nature of lapse and shift in this framework (see (2.24)), the (geometry parts of the) diffeomorphism constraints C i must fall off strictly faster than the (geometry parts of the) Hamiltonian constraint C in order to accommodate asymptotically flat spacetimes:

So one has to be careful in choosing elementary fields Q ij , P ij such that these conditions hold. As we have seen in the last section the standard conditions are generally not strong enough for this purpose, they only guarantee that C → O(σ r -(1+ ) ) and C σr → O(σ r -( 1+) ). In this work we did satisfy this stronger fall off behaviour by demanding that the shift (or equivalently the momentum density) N i = -Q ij C j /H = 0 vanishes. It is then sufficient to add to the physical Hamiltonian the ADM mass term

In the general case one has to choose dynamical fields Λ, R, P Λ , P R such that, in addition to (A.3), also (A.4) holds. This means that the physical Hamiltonian decays as H → O(σ r -(1+ ) ) and asymptotically lapse and shift behave as

B Spherically Symmetric Coordinate Systems

In the main text we have worked with various presentations of spherically symmetric metrics in various coordinate systems. For the benefit of the reader we recall here how these coordinate systems are related with each other. Our notation is as follows: We call (τ, σ r ) the Lemaitre time and radial coordinate which coincide