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Talk:Armenian Soviet Socialist Republic

translation
A translation is welcome, I cannot read it. If I cant read it, it is useless as no translation exist.

POV in history section
the history section makes it look like Armenia was independent up until 1920. it spent 3 years as a weak and unrecognized state, and was previously part of the Russian empire. however the history section here makes it seem like it was a well-established and fully sovereign nation and the soviet union was a big mean aggressive empire that gobbled up weaker nations like the roman empire did. could someone fix this? Pure inuyasha 23:27, 6 June 2006 (UTC)
 * Even if they were part of Russian Empire they were all recognized by the League of Nations-proto UN as independent states.Cheers! Armenian-nj
 * Actually, Armenia (as well as its neighbors - Georgia and Azerbaijan) were all well-established, fully sovereign and recognized republics. Unfortunately, though their freedom did not last and they were invaded (without any major resistance) by the Bolsheviks who proclaimed them Soviet republics.  The same story comes up in books such as The Complete Guide to the Soviet Union and Armenia: Past and Present (both cited), so it is true. -- Clevelander 00:42, 7 June 2006 (UTC)

What i mean is that there should be some mention of it being part of the Russian empire. Pure inuyasha 14:27, 7 June 2006 (UTC)

oh, and i don't belive they were recognized, i have a 1920 atlas that still shows it as part of the russian empire. i don't see how it could be regocnized when the country it broke off of was still recognized! Pure inuyasha 14:28, 7 June 2006 (UTC)

 * How can an unrecognized state have ambassadors from US, France and GB? The Treaty of Sevres made all signatories recognise Armenia as a sovereign state! There was no such thing as Russia since 1917.--Eupator 18:19, 7 June 2006 (UTC)

I know it was recognised! What I'm saying is that there should be some sign that it was part of the russian empire up to 1917. Pure inuyasha 21:32, 22 June 2006 (UTC)

The "Armenian Legion"
I believe it should be included in the section under "Great Patriotic War" that a number of Armenians served in German military units. This is a known fact which has been documented by serveral sources. Its really no different thatn the Russians, Ukrainians, Latvians or Central Asians who also served in similar unites (such as the so called Russian Liberation Army of Andrei Vlasov) Piercetp 00:04, 15 June 2006 (UTC) Can you prove anyhow that Armenians served in Nazi Army? I can prove the opposite-Misak Manushian -the leader of French resistance was Armenian.
 * I included this earlier (along with information on how some even helped save Jews from the Holocaust while part of it). I re-added my earlier paragraph for now, but I'm still not sure if we should include it on an article about the Armenian SSR specfically.  Perhaps if there was an article called "Armenians in World War II" then we could do it.  Again, I'm not sure.  -- Clevelander 02:18, 15 June 2006 (UTC)
 * Er, also, I'm willing to discuss topics such as these, but I prefer that you do not accuse me of "vandalism." Remember, I have worked long and hard on this article too. Cheers, Clevelander 02:20, 15 June 2006 (UTC)
 * OK that works. Thanks. About that "vandalism", I actually posted a second time because I believed that was too strong a word. I tried to delete it as soon as I posted it but there is no way to reedit comment summaries here. Sorry. Piercetp 04:05, 15 June 2006 (UTC)
 * No problem! -- Clevelander 22:26, 15 June 2006 (UTC)

GA on hold
I think this article is close to being a Good Article, but a few things remain, so I am putting it on hold and if these issues are fixed within the next week then this article can be promoted.

 * Unfortunately the images are problematic and probably need to be removed:
 * Image:SchoolchildrenInArmenia.jpg does not qualify under fair use. It would be fair use if you were writing about that specific photo, not what is illustrating.
 * Image:May1paradeYerevan.jpg is from the Armenian Soviet Encyclopedia which was published in the 70s/80s and hence is not Public Domain.
 * I have spotted a bunch of grammatical and spelling problems with the aritcle. I have fixed up a couple, but the article still needs a good copyedit to check the grammar.  Also, there are two mixes of English here - most words are spelt using U.S. English, but some words are spelt in U.K. English, just a couple of instances "metres", "civilisation" (UK) "neighbors", "theaters" (US)  (though this is a minor issue).--Konstable 08:38, 20 February 2007 (UTC)

 * One thing which is still a bit confusing: "Under the Soviet system, the economy of the country was highly centralized with little to no private ownership of income producing property." I would fix this myself but I really don't understand what it's saying in the first place.--Konstable 02:24, 22 February 2007 (UTC)

 * Alright I changed it, basically, what it's saying is that communism took over and capitalism was out of the window. The economy was centralized, controlled by the government so they banned all privately runned businesses. - Fedayee 02:42, 22 February 2007 (UTC)

 * I think there is still a lot to be done here. It is not just a problem of grammar (still some oversized sentences that are hard to read here and there).  A lot of the time one sentence does not logically follow another, and some key dates are left out.  I will see if I can help fix some of these myself, though not sure if I will have the time.--Konstable 13:05, 22 February 2007 (UTC)

 * I think the current version is good enough to be promoted to GA status. I would do this myself right now, but I think I might have worked too much on this article to qualify as a reviewer for it.  I will leave a note on WP:GAC for someone else to confirm that it should be promoted.--Konstable 04:59, 26 February 2007 (UTC)

Date: 1936?
Any reason why Armenian SSR's existence dates in the box is written as 1936-1991? Officially, the Republic was established in December of 1920, right? Why do you refer to the disintegration of Union of Soviet Transcaucasian Republics as the establishment of Armenian SSR? Thanks for the explanation in advance. --Aynabend (talk) 14:55, 25 December 2007 (UTC)

Proposed move to Armenian Soviet Socialist Republic
It seems that given that the formal name for this nation was "Armenian Soviet Socialist Republic", the soviet prefix abbreviation (-"SSR") shouldn't be in the full article name. Therefore, I propose to move the article to its full form, Armenian Soviet Socialist Republic. We wouldn't have an article about North Korea titled "DPRK", would we? --Micahbrwn (talk) 09:54, 29 December 2007 (UTC)
 * Fine by me, but what about other 15 republics? They should all be named the same way. VartanM (talk) 18:46, 29 December 2007 (UTC)
 * Please discuss this multimove here --Lox (t,c) 11:48, 30 December 2007 (UTC)

culture and life
is the flag anecdote really relevant here? it seems to be included more as a putdown towards turkey in the ongoing saga of turkish-armenian animosity, more than anything else. not to mention the only disputable source:...krutchev's memoires —Preceding unsigned comment added by 78.29.210.205 (talk) 06:35, 7 May 2008 (UTC)

More improvement needed
The article, despite its GA status, contains numerous inaccuracies, weasel-worded statements, and unsourced assertions. Also, given the importance of the subject, the article should rely on more reliable and up-to-date sources. --KoberTalk 06:53, 9 March 2015 (UTC)

 * I concur. Its GA status was given many years ago though, so perhaps the quality standards back then were somewhat lower. Not something that can't be solved though. - LouisAragon (talk) 23:49, 8 December 2015 (UTC)

Article editing
There are certain users who have been editing other SSR articles on Wikipedia for the past year, by stating that the soviet republics existed until the adoption of their new constitutions in the mid-1990s.

Armenian SSR declared itself independent and changed its name to Armenia in 1991, and there are absolutely no arguments to back up that this state existed until 1995. That would be rewriting history. A new state is not just a matter of all-new constitutions, but also of its status and form of government. Afghanistan has had a lot of states since the abolishment of monarchy in the 1970s: the First Republic (1973–1978), Democratic Republic/Second Republic (1978–1992), the Islamic State (1992–2001), the Islamic Emirate (1996–2001), the Afghan Interim Administration (2001–2002), the Afghan Transitional Administration (2002–2004), and the present-day Islamic Republic since 2004. Yet they have only had four constitutions since then: 1976, 1987, 1990 and 2004. Is that to say that we should change these yearspans totally as well, so that most of these states didn't exist? The People's Republic of Hungary ended in 1989, but an all-new constitution was first made in 2011. Should we also say that the PR of Hungary existed until 2011, then? That would make little sense. Although Armenia did not adopt a new constitution before 1995, there's no doubt about that it was a totally different state. It had a totally different form of government, its name was changed, it was an independent state (not a federated state), it was not a Soviet socialist republic. These factors are a lot more independent than the adoption of an all-new constitution. And although the constitution was not all-new, and formally the same constitution although heavily amended, it was amended to fit a new state and was not really the same constitution in practice. You'll have to agree that the 1991 transition is a lot more historically significant change in Armenia's history than the adoption of a new constitution.  A.h. king  • Talk to me!  22:03, 24 February 2017 (UTC)

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Fashion Show Mall

Fashion Show is a shopping mall located on the Las Vegas Strip in Paradise, Nevada. It was developed by Summa Corporation and Ernest W. Hahn, the latter also serving as general contractor. The mall features Dick's Sporting Goods, Dillard's, Macy's, Macy's Men's Store, Nordstrom, Neiman Marcus, and Saks Fifth Avenue.

Construction began on March 15, 1979, and the project was built at a cost of $74 million. Fashion Show Mall opened on February 14, 1981, becoming the first major shopping center on the Strip. It was also the third mall to open in the Las Vegas Valley. Its success prompted the construction of other retail centers on the Strip, including the Forum Shops.

The Rouse Company took over ownership in the 1990s and later conducted a $1 billion renovation and expansion, which lasted from 2000 to 2003. The project included new stores and restaurants, as well as a retractable runway for fashion shows. A plaza along the Strip was also added and includes the Cloud, a steel structure providing shade to pedestrians, while doubling as a nighttime projection surface for advertising. General Growth Properties (now Brookfield Properties) acquired the mall in 2004.

History
Summa Corporation announced Fashion Show Mall in July 1978, to be built on 34 acre on the Las Vegas Strip, south of Summa's Frontier hotel-casino. It would be co-developed with Ernest W. Hahn, which would also serve as general contractor. It was designed by ArchiSystems, based in Van Nuys, California. The project had been planned for years. The Frontier's Little Church of the West occupied a portion of the future mall site, and was relocated elsewhere on the Strip. Groundbreaking took place on March 15, 1979.

Built at a cost of $74 million, Fashion Show Mall opened on February 14, 1981. It was the first major shopping center to open on the Strip, and the third to open in the Las Vegas Valley. Tourists were the primary demographic. The mall was 75-percent owned by Summa, while Hahn held the remaining ownership. The mall was successful, prompting the addition of other retail centers on the Strip, including the Forum Shops. A $10 million renovation was completed in 1993, after three years of planning and construction.

Summa later became the Howard Hughes Corporation, and was sold to The Rouse Company in 1996. A year later, Rouse announced plans to expand Fashion Show Mall, while new management began hosting events there to attract additional patrons. In 1998, Rouse purchased 15 acres of property just west of the mall, previously used by Mirage Resorts as an employee parking lot. Rouse bought out Hahn's ownership stake a month later.

In May 2000, work began on an expansion and remodel,  a project that tenants felt was overdue. A new west wing, the first phase of the $1 billion expansion project, opened on November 1, 2002. It introduced a retractable runway and stage used for fashion shows. The space has also been used to put on Christmas shows.

The existing east wing debuted on October 1, 2003, following renovations. The expansion and renovation helped the mall compete against newer retail centers on the Strip, such as the Forum Shops and the Grand Canal Shoppes, both of which were also in the process of expansion. General Growth Properties (now Brookfield Properties) acquired the mall in 2004, when it purchased Rouse.

Various incidents have occurred at the mall, including an unsuccessful bomb threat in 1983, and a jewelry store heist in 1999. Several shootings have occurred, including one at the food court in 2014. Another shooting in 2019 prompted the evacuation of the mall. In 2022, a woman was shot and killed in the mall's parking garage during an attempted robbery.

In 2014, the mall was used to film the interior shots of the fictitious West Orange Pavilion Mall for the opening sequence of Paul Blart: Mall Cop 2.

Stores
Fashion Show Mall opened with 822700 sqft. Anchor tenants occupied 500000 sqft, and included Bullock's, Diamond's, Goldwater's, Neiman-Marcus, and Saks Fifth Avenue. The latter two introduced luxury designer brands to Las Vegas. Aside from anchors, the mall had 130 tenant spaces, and opened with 87 stores. Diamond's was converted to a Dillard's store in 1984. The first Bullock's Woman store opened at Fashion Show Mall in 1987. Goldwater's became a May Company store in 1989 and then a Robinsons-May in 1993, while Bullock's was converted to a Macy's in 1996.

The expansion in the early 2000s doubled the mall's size to approximately 1900000 sqft. This made it one of the largest malls in the U.S., while surpassing The Boulevard Mall as the largest in Las Vegas. Notable new tenants in the expansion included Abercrombie & Fitch, Ann Taylor Loft, Claire's, Guess, and the first Apple Store in Las Vegas. Other tenants added since then have included Banana Republic, Henri Bendel, and Disney Store.

Mall profits had been held back by the limited size of its anchor stores, which were enlarged during the expansion. Dillard's relocated to a 200000 sqft space in the new west wing. Saks Fifth Avenue occupied a new 170000 sqft addition, marking its largest location outside of New York. The expansion project also added two new anchor tenants for a total of seven, more than any other mall in the U.S., surpassing the five-anchor Mall of America. The new anchors included Nordstrom, making its Las Vegas debut; and the first Bloomingdale's Home location. An eighth anchor, Lord & Taylor, was scheduled to take the former Dillard's space. However, parent company May Department Stores dropped these plans, citing poor economic conditions. The vacant space was eventually taken over by Forever 21, which opened in 2010. Robinsons-May eventually closed, and was replaced in 2013 by a Macy's Men store. Bloomingdale's store closed that year, and was replaced two years later by Dick's Sporting Goods.

As of 2014, the mall had 250 stores.

Plaza and Cloud
A plaza, measuring 72000 sqft, opened at the east entrance in 2003, as part of the expansion. The project also added the Cloud, an oval-shaped structure located above the plaza. It provides shade during the day, while its underside serves as a nighttime projection screen for paid advertisers.

Construction of the Cloud took more than a year. The structure weighs 1.1 million pounds, and is 479 feet long and 160 feet wide. Situated at an angle, its lowest and highest point above the ground is 96 and 128 feet respectively. The Cloud is stationed in place with the use of cables, preventing strong winds from blowing it away. The structure is mostly made of steel, which was prefabricated and assembled on-site before being lifted into place. Construction took place along Las Vegas Boulevard, and the safety of nearby traffic posed the biggest construction challenge. The plaza was revamped in 2015 to include new LED screens.

Restaurants
Approximately 46000 sqft of restaurant space was added in the early 1990s. Dive!, a sea-themed restaurant, opened in 1996. It was a joint venture between film director Steven Spielberg and businessmen Steve Wynn and Jeffrey Katzenberg. Dive! was demolished in 2001, making way for the plaza area.

California Pizza Kitchen was among several new restaurants opened in 2002, during the first phase of expansion. Additional restaurants opened in 2004, including The Capital Grille, Maggiano's Little Italy, and RA Sushi. The original food court was removed, and a new one opened in the renovated east wing, measuring 11288 sqft. It occupies a third floor overlooking the Strip. As of 2014, the mall had 15 dining establishments. Plans were announced that year to add five more restaurants to the plaza area. In 2018, chef Tom Douglas opened an Italian restaurant named Jeannie's. A Hello Kitty-themed cafe opened in 2022. Former football player Emmitt Smith is scheduled to open a restaurant, Emmitt Las Vegas, by early 2023.

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Talk:Dunhill

I'm not sure how to fix this. There's nothing here right now, but it's misleading-- there is no Dunhill Tobacco Company. Dunhill, with the death of the founder Alfred Dunhill, was split into a pipe company (which was retained by the Dunhill family), a luxury goods company (bought by a South African firm I need to look up the name of) and the tobacco rights were sold to BAT (British American Tobacco). Until now (until, well, the end of April 2005), Dunhill tobaccos have been made by Murray's in Belfast (a pipe tobacco factory owned by BAT). Some of the Dunhill tobaccos, including Dunhill Navy Rolls (which is on the European market only-- it's complicated based on intellectual property owned by Orlik of Denmark, which owns A&C Petersen, which owned rights in the US to Escudo, but EU IP remained questionable-- anyway, as I said, it's complicated. Orlik is half owned by BAT also) are made at the Orlik factory in Denmark, where production for the others (now that Murray's in Belfast is closing) will likely go. Anyway, the short and long of it is that that there is no Dunhill tobacco company. I could write the entry on Dunhill from its origins at the beginning of the 20th century until today, but I'm, well, a bit confused as to taxonomy and structure here and would appreciate a push in the right direction from someone who is a little more experienced with building Wikipedia entries from scratch.

Ugh. European tobacco industry incestuousness managed to confuse that paragraph. The point is that there is no Dunhill Tobacco Co. and that the Dunhill pipe tobaccos (among the best selling pipe tobaccos worldwide) are not owned by Dunhill, but rather are owned by BAT and made in both Northern Ireland (until the end of this month) and Denmark.

To be precise, there is a small production of hand blended tobacco at Dunhill's. It could be bought only in Alfred Dunhill shop in London.

Sadly, dating the end of 2005, London Dunhill shop doesn't sell tobacco products anymore.

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Ram-air intake

A ram-air intake is any intake design which uses the dynamic air pressure created by vehicle motion, or ram pressure, to increase the static air pressure inside of the intake manifold on an internal combustion engine, thus allowing a greater massflow through the engine and hence increasing engine power.

Design features

The ram-air intake works by reducing the intake air velocity by increasing the cross-sectional area of the intake ducting. When gas velocity goes the pressure is increased. The increased pressure in the air box will ultimately have a positive effect on engine output as more oxygen will enter the cylinder during each engine cycle.

Ram-air systems are used on high-performance vehicles, most often on motorcycles and performance cars. The 1990 Kawasaki Ninja ZX-11 C1 model used a ram-air intake, the very first on any production motorcycle. Ram-air was a feature on some cars in the sixties. It fell out of favor in the seventies, but recently made a comeback. While ram-air intakes may increase the volumetric efficiency of an engine, they can be difficult to combine with carburetors, which rely on a venturi-engineered pressure drop to draw fuel through the main jet. As the pressurised ram-air may kill this venturi effect, the carburetor needs to be designed to take this into account, or, alternatively, the engine may need fuel-injection.

At low speeds (subsonic speeds) increases in static pressure are however limited to a few percent.

Aircraft
Pitot sensors are used to measure ram pressure which, along with static pressure, is used to estimate the airspeed of an aircraft.

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Four-gradient

In differential geometry, the four-gradient (or 4-gradient) $$\boldsymbol{\partial}$$ is the four-vector analogue of the gradient $$\vec{\boldsymbol{\nabla}}$$ from vector calculus.

In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.

Notation
This article uses the $(+ − − −)$ metric signature.

SR and GR are abbreviations for special relativity and general relativity respectively.

$$c$$ indicates the speed of light in vacuum.

$$\eta_{\mu\nu} = \operatorname{diag}[1,-1,-1,-1]$$ is the flat spacetime metric of SR.

There are alternate ways of writing four-vector expressions in physics:
 * The four-vector style can be used: $$\mathbf{A} \cdot \mathbf{B}$$, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. $$\vec{\mathbf{a}} \cdot \vec{\mathbf{b}}$$. Most of the 3-space vector rules have analogues in four-vector mathematics.
 * The Ricci calculus style can be used: $$A^\mu \eta_{\mu\nu} B^\nu$$, which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as $$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$$.

The Latin tensor index ranges in {1, 2, 3}, and represents a 3-space vector, e.g. $$A^i = \left(a^1, a^2, a^3\right) = \vec{\mathbf{a}}$$.

The Greek tensor index ranges in {0, 1, 2, 3}, and represents a 4-vector, e.g. $$A^\mu = \left(a^0, a^1, a^2, a^3\right) = \mathbf{A}$$.

In SR physics, one typically uses a concise blend, e.g. $$\mathbf{A} = \left(a^0, \vec{\mathbf{a}}\right)$$, where $$a^0$$ represents the temporal component and $$\vec{\mathbf{a}}$$ represents the spatial 3-component.

Tensors in SR are typically 4D $$(m,n)$$-tensors, with $$m$$ upper indices and $$n$$ lower indices, with the 4D indicating 4 dimensions = the number of values each index can take.

The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation): $$\mathbf{A} \cdot \mathbf{B} = A^\mu \eta_{\mu\nu} B^\nu = A_\nu B^\nu = A^\mu B_\mu = \sum_{\mu=0}^{3} a^\mu b_\mu = a^0 b^0 - \sum_{i=1}^{3} a^i b^i = a^0 b^0 - \vec{\mathbf{a}} \cdot \vec{\mathbf{b}}$$

Definition
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are: $$\dfrac{\partial}{\partial X^\mu} = \left(\partial_0,\partial_1,\partial_2,\partial_3\right) = \left(\partial_0,\partial_i\right) = \left(\frac{1}{c}\frac{\partial}{\partial t}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \partial_x,\partial_y,\partial_z\right) = \partial_\mu = {}_{,\mu}$$

The comma in the last part above $${}_{,\mu}$$ implies the partial differentiation with respect to 4-position $$X^\mu$$.

The contravariant components are: $$\boldsymbol{\partial} = \partial^\alpha = \eta^{\alpha \beta} \partial_\beta = \left(\partial^0,\partial^1,\partial^2,\partial^3\right) = \left(\partial^0,\partial^i\right) = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right) = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \left(\frac{\partial_t}{c}, -\partial_x,-\partial_y,-\partial_z\right)$$

Alternative symbols to $$\partial_\alpha$$ are $$\Box$$ and D (although $$\Box$$ can also signify $$\partial^\mu \partial_\mu$$ as the d'Alembert operator).

In GR, one must use the more general metric tensor $$g^{\alpha \beta}$$ and the tensor covariant derivative $$\nabla_{\mu} = {}_{;\mu}$$ (not to be confused with the vector 3-gradient $$\vec{\nabla}$$).

The covariant derivative $$\nabla_{\nu}$$ incorporates the 4-gradient $$\partial_\nu$$ plus spacetime curvature effects via the Christoffel symbols $$ \Gamma^{\mu}{}_{\sigma \nu} $$

The strong equivalence principle can be stated as:

"Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas in SR are simply changed to covariant derivative semi-colons  in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule".

So, for example, if $$T^{\mu\nu}{}_{,\mu} = 0$$ in SR, then $$T^{\mu\nu}{}_{;\mu} = 0$$ in GR.

On a (1,0)-tensor or 4-vector this would be: $$\begin{align} \nabla_\beta V^\alpha &= \partial_\beta V^\alpha + V^\mu \Gamma^{\alpha}{}_{\mu\beta} \\[0.1ex] V^{\alpha}{}_{ ;\beta} &= V^{\alpha}{}_{ ,\beta} + V^\mu \Gamma^{\alpha}{}_{\mu\beta} \end{align}$$

On a (2,0)-tensor this would be: $$\begin{align} \nabla_{\nu} T^{\mu \nu} &= \partial_\nu T^{\mu \nu} + \Gamma^{\mu}{}_{\sigma \nu}T^{\sigma \nu} + \Gamma^{\nu}{}_{\sigma \nu} T^{\mu \sigma} \\ T^{\mu \nu}{}_{;\nu} &= T^{\mu \nu}{}_{,\nu} + \Gamma^{\mu}{}_{\sigma \nu}T^{\sigma \nu} + \Gamma^{\nu}{}_{\sigma \nu} T^{\mu \sigma} \end{align}$$

Usage
The 4-gradient is used in a number of different ways in special relativity (SR):

Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of SR, but have to be modified for the more general curved space coordinates of general relativity (GR).

As a 4-divergence and source of conservation laws
Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. Note that in this metric signature [+,−,−,−] the 4-Gradient has a negative spatial component. It gets canceled when taking the 4D dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1].

The 4-divergence of the 4-position $$X^\mu = \left(ct, \vec{\mathbf{x}}\right)$$ gives the dimension of spacetime: $$\boldsymbol{\partial} \cdot \mathbf{X} = \partial^\mu \eta_{\mu\nu} X^\nu = \partial_\nu X^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot (ct,\vec{x}) = \frac{\partial_t}{c}(ct) + \vec{\nabla}\cdot \vec{x} = (\partial_t t) + (\partial_x x + \partial_y y + \partial_z z) = (1) + (3) = 4$$

The 4-divergence of the 4-current density $$J^\mu = \left(\rho c, \vec{\mathbf{j}}\right) = \rho_o U^\mu = \rho_o \gamma\left(c, \vec{\mathbf{u}}\right) = \left(\rho c, \rho \vec{\mathbf{u}}\right)$$ gives a conservation law – the conservation of charge: $$\boldsymbol{\partial} \cdot \mathbf{J} = \partial^\mu \eta_{\mu\nu} J^\nu = \partial_\nu J^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot (\rho c,\vec{j}) = \frac{\partial_t}{c} (\rho c) + \vec{\nabla} \cdot \vec{j} = \partial_t \rho + \vec{\nabla} \cdot \vec{j} = 0$$

This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density $$\partial_t \rho = -\vec{\nabla}\cdot \vec{j}$$.

In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.

The 4-divergence of the 4-number flux (4-dust) $$N^\mu = \left(nc, \vec{\mathbf{n}}\right) = n_o U^\mu = n_o \gamma\left(c, \vec{\mathbf{u}}\right) = \left(nc, n\vec{\mathbf{u}}\right)$$ is used in particle conservation: $$\boldsymbol{\partial} \cdot \mathbf{N} = \partial^\mu \eta_{\mu\nu} N^\nu = \partial_\nu N^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot \left(nc, n\vec{\mathbf{u}}\right) = \frac{\partial_t}{c} \left(nc\right) + \vec{\nabla} \cdot n \vec{\mathbf{u}} = \partial_t n + \vec{\nabla}\cdot n\vec{\mathbf{u}} = 0$$

This is a conservation law for the particle number density, typically something like baryon number density.

The 4-divergence of the electromagnetic 4-potential $A^\mu = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right)$ is used in the Lorenz gauge condition: $$\boldsymbol{\partial} \cdot \mathbf{A} = \partial^\mu \eta_{\mu\nu} A^\nu = \partial_\nu A^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot \left(\frac{\phi}{c}, \vec{a}\right) = \frac{\partial_t}{c} \left(\frac{\phi}{c}\right) + \vec{\nabla} \cdot \vec{a} = \frac{\partial_t \phi}{c^2} + \vec{\nabla} \cdot \vec{a} = 0$$

This is the equivalent of a conservation law for the EM 4-potential.

The 4-divergence of the transverse traceless 4D (2,0)-tensor $$h^{\mu\nu}_{TT}$$ representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).

The transverse condition $$\boldsymbol{\partial} \cdot h^{\mu\nu}_{TT} = \partial_\mu h^{\mu\nu}_{TT} = 0$$ is the equivalent of a conservation equation for freely propagating gravitational waves.

The 4-divergence of the stress–energy tensor $$T^{\mu \nu}$$ as the conserved Noether current associated with spacetime translations, gives four conservation laws in SR:

The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions). $$\boldsymbol{\partial} \cdot T^{\mu \nu} = \partial_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} = 0^\mu = (0,0,0,0)$$

It is often written as: $$\partial_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} = 0$$ where it is understood that the single zero is actually a 4-vector zero $$0^\mu = (0,0,0,0)$$.

When the conservation of the stress–energy tensor ($\partial_{\nu} T^{\mu \nu} = 0^\mu $) for a perfect fluid is combined with the conservation of particle number density ($$\boldsymbol{\partial} \cdot \mathbf{N} = 0$$), both utilizing the 4-gradient, one can derive the relativistic Euler equations, which in fluid mechanics and astrophysics are a generalization of the Euler equations that account for the effects of special relativity. These equations reduce to the classical Euler equations if the fluid 3-space velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.

In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum (relativistic angular momentum) is also conserved: $$\partial_\nu \left(x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu}\right) = \left(x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu}\right)_{,\nu} = 0^{\alpha \mu}$$ where this zero is actually a (2,0)-tensor zero.

As a Jacobian matrix for the SR Minkowski metric tensor
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

The 4-gradient $$\partial^\mu$$ acting on the 4-position $$X^\nu$$ gives the SR Minkowski space metric $$\eta^{\mu\nu}$$: $$\begin{align} \boldsymbol{\partial} [\mathbf{X}] = \partial^\mu[X^\nu] = X^{\nu_,\mu} &= \left(\frac{\partial_t}{c}, -\vec{\nabla}\right)\left[\left(ct, \vec{x}\right)\right] = \left(\frac{\partial_t}{c}, -\partial_x, -\partial_y, -\partial_z\right)[(ct, x, y, z)], \\[3pt] &= \begin{bmatrix} \frac{\partial_t}{c} ct & \frac{\partial_t}{c} x & \frac{\partial_t}{c} y & \frac{\partial_t}{c} z \\ -\partial_x ct & -\partial_x x & -\partial_x y & -\partial_x z \\ -\partial_y ct & -\partial_y x & -\partial_y y & -\partial_y z \\ -\partial_z ct & -\partial_z x & -\partial_z y & -\partial_z z      \end{bmatrix} = \begin{bmatrix} 1 & 0 &  0 &  0 \\         0 & -1 &  0 &  0 \\         0 &  0 & -1 &  0 \\         0 &  0 &  0 & -1       \end{bmatrix} \\[3pt] &= \operatorname{diag}[1,-1,-1,-1] = \eta^{\mu\nu}. \end{align}$$

For the Minkowski metric, the components $$\left[\eta^{\mu\mu}\right] = 1/\left[\eta_{\mu\mu}\right]$$ ($$\mu$$ not summed), with non-diagonal components all zero.

For the Cartesian Minkowski Metric, this gives $$\eta^{\mu\nu} = \eta_{\mu\nu} = \operatorname{diag}[1, -1, -1, -1]$$.

Generally, $$\eta_\mu^\nu = \delta_\mu^\nu = \operatorname{diag}[1,1,1,1]$$, where $$\delta_\mu^\nu$$ is the 4D Kronecker delta.

As a way to define the Lorentz transformations
The Lorentz transformation is written in tensor form as $$X^{\mu'} = \Lambda^{~\mu'}_{\nu} X^\nu$$ and since $$\Lambda^{\mu'}_\nu$$ are just constants, then $$\dfrac{\partial X^{\mu'} }{ \partial X^\nu} = \Lambda^{\mu'}_\nu$$

Thus, by definition of the 4-gradient $$ \partial_\nu \left[X^{\mu'}\right] = \left(\dfrac{\partial}{\partial X^\nu}\right)\left[X^{\mu'}\right] = \dfrac{\partial X^{\mu'}}{\partial X^\nu} = \Lambda^{\mu'}_\nu $$

This identity is fundamental. Components of the 4-gradient transform according to the inverse of the components of 4-vectors. So the 4-gradient is the "archetypal" one-form.

As part of the total proper time derivative
The scalar product of 4-velocity $$U^\mu$$ with the 4-gradient gives the total derivative with respect to proper time $$\frac{d}{d\tau}$$: $$\begin{align} \mathbf{U} \cdot \boldsymbol{\partial} &= U^\mu \eta_{\mu\nu} \partial^\nu = \gamma \left(c, \vec{u}\right) \cdot \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \gamma \left(c \frac{\partial_t}{c} + \vec{u} \cdot \vec{\nabla} \right) = \gamma \left(\partial_t + \frac{dx}{dt} \partial_x + \frac{dy}{dt} \partial_y + \frac{dz}{dt} \partial_z \right) = \gamma \frac{d}{dt} = \frac{d}{d\tau} \\ \frac{d}{d\tau} &= \frac{dX^\mu}{dX^\mu} \frac{d}{d\tau} = \frac{dX^\mu}{d\tau} \frac{d}{dX^\mu} = U^\mu \partial_\mu = \mathbf{U} \cdot \boldsymbol{\partial} \end{align}$$

The fact that $$\mathbf{U} \cdot \boldsymbol{\partial}$$ is a Lorentz scalar invariant shows that the total derivative with respect to proper time $$\frac{d}{d\tau}$$ is likewise a Lorentz scalar invariant.

So, for example, the 4-velocity $$U^\mu$$ is the derivative of the 4-position $$X^\mu$$ with respect to proper time: $$\frac{d}{d\tau} \mathbf{X} = (\mathbf{U} \cdot \boldsymbol{\partial})\mathbf{X} = \mathbf{U} \cdot \boldsymbol{\partial}[\mathbf{X}] = U^\alpha \cdot \eta^{\mu\nu} = U^\alpha \eta_{\alpha \nu} \eta^{\mu\nu} = U^\alpha \delta_\alpha^\mu = U^\mu = \mathbf{U} $$ or $$\frac{d}{d\tau} \mathbf{X} = \gamma\frac{d}{dt} \mathbf{X} = \gamma\frac{d}{dt} \left(ct, \vec{x}\right) = \gamma \left(\frac{d}{dt}ct,\frac{d}{dt}\vec{x}\right) = \gamma \left(c, \vec{u}\right) = \mathbf{U}$$

Another example, the 4-acceleration $$A^\mu$$ is the proper-time derivative of the 4-velocity $$U^\mu$$: $$\begin{align} \frac{d}{d\tau} \mathbf{U} &= (\mathbf{U} \cdot \boldsymbol{\partial})\mathbf{U} = \mathbf{U} \cdot \boldsymbol{\partial}[\mathbf{U}] = U^\alpha \eta_{\alpha\mu}\partial^\mu\left[U^\nu\right] \\ &= U^\alpha \eta_{\alpha\mu}\begin{bmatrix} \frac{\partial_t}{c} \gamma c & \frac{\partial_t}{c} \gamma \vec{u} \\ -\vec{\nabla}\gamma c & -\vec{\nabla}\gamma \vec{u} \end{bmatrix} = U^\alpha \begin{bmatrix}\ \frac{\partial_t}{c} \gamma c & 0 \\ 0 & \vec{\nabla}\gamma \vec{u} \end{bmatrix} \\[3pt] &= \gamma \left(c \frac{\partial_t}{c} \gamma c, \vec{u} \cdot \nabla\gamma \vec{u}\right) = \gamma \left(c \partial_t \gamma, \frac{d}{dt}\left[\gamma \vec{u}\right]\right) = \gamma \left(c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}}\right) = \mathbf{A} \end{align}$$

or $$\frac{d}{d\tau} \mathbf{U} = \gamma \frac{d}{dt} (\gamma c,\gamma \vec{u}) =\gamma \left(\frac{d}{dt} [\gamma c],\frac{d}{dt} [\gamma \vec{u}] \right) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} ) = \mathbf{A} $$

As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations
The Faraday electromagnetic tensor $$F^{\mu\nu}$$ is a mathematical object that describes the electromagnetic field in spacetime of a physical system.

Applying the 4-gradient to make an antisymmetric tensor, one gets: $$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu = \begin{bmatrix} 0    & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0     & -B_z   & B_y    \\ E_y/c & B_z   & 0      & -B_x   \\ E_z/c & -B_y  & B_x    & 0 \end{bmatrix} $$ where:
 * Electromagnetic 4-potential $$A^\mu = \mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right)$$, not to be confused with the 4-acceleration $$\mathbf{A} = \gamma \left(c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}}\right)$$
 * The electric scalar potential is $$\phi$$
 * The magnetic 3-space vector potential is $$\vec{\mathbf{a}}$$

By applying the 4-gradient again, and defining the 4-current density as $$J^{\beta} = \mathbf{J} = \left(c\rho, \vec{\mathbf{j}}\right)$$ one can derive the tensor form of the Maxwell equations: $$\partial_{\alpha} F^{\alpha\beta} = \mu_o J^{\beta}$$ $$\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0_{\alpha \beta \gamma}$$ where the second line is a version of the Bianchi identity (Jacobi identity).

As a way to define the 4-wavevector
A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation

The 4-wavevector $$K^\mu$$ is the 4-gradient of the negative phase $$\Phi$$ (or the negative 4-gradient of the phase) of a wave in Minkowski Space: $$K^\mu = \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \boldsymbol{\partial} [-\Phi] = -\boldsymbol{\partial} [\Phi]$$

This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave): $$\mathbf{K} \cdot \mathbf{X} = \omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}} = -\Phi$$

where 4-position $$\mathbf{X} = \left(ct, \vec{\mathbf{x}}\right)$$, $$\omega$$ is the temporal angular frequency, $$\vec{\mathbf{k}}$$ is the spatial 3-space wavevector, and $$\Phi$$ is the Lorentz scalar invariant phase.

$$\partial [\mathbf{K} \cdot \mathbf{X}] = \partial \left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right] = \left(\frac{\partial_t}{c}, -\nabla\right)\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right] = \left(\frac{\partial_t}{c}\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right], -\nabla\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right]\right) = \left(\frac{\partial_t}{c}[\omega t], -\nabla\left[- \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right]\right) = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \mathbf{K} $$ with the assumption that the plane wave $$\omega$$ and $$\vec{\mathbf{k}}$$ are not explicit functions of $$t$$ or $$\vec{\mathbf{x}}$$.

The explicit form of an SR plane wave $$\Psi_n(\mathbf{X})$$ can be written as:

$$\Psi_n(\mathbf{X}) = A_ne^{-i(\mathbf{K_n}\cdot\mathbf{X})} = A_ne^{i(\Phi_n)}$$ where $$A_n$$ is a (possibly complex) amplitude.

A general wave $$\Psi(\mathbf{X})$$ would be the superposition of multiple plane waves: $$\Psi(\mathbf{X}) = \sum_{ n }[\Psi_n(\mathbf{X})] = \sum_{ n }\left[A_{n} e^{-i(\mathbf{K_n}\cdot \mathbf{X})}\right] = \sum_{ n }\left[A_{n} e^{i(\Phi_n)}\right]$$

Again using the 4-gradient, $$\partial [\Psi(\mathbf{X})] = \partial\left[Ae^{-i(\mathbf{K}\cdot\mathbf{X})}\right] = -i\mathbf{K} \left[Ae^{-i(\mathbf{K}\cdot\mathbf{X})}\right] = -i\mathbf{K} [\Psi(\mathbf{X})] $$ or $$\boldsymbol{\partial} = -i \mathbf{K}$$ which is the 4-gradient version of complex-valued plane waves

As the d'Alembertian operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator, also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.

The square of $$\boldsymbol{\partial}$$ is the 4-Laplacian, which is called the d'Alembert operator:

$$\boldsymbol{\partial} \cdot \boldsymbol{\partial} = \partial^\mu \cdot \partial^\nu = \partial^\mu \eta_{\mu\nu} \partial^\nu = \partial_\nu \partial^\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \vec{\nabla}^2 = \left(\frac{\partial_t}{c}\right)^2 - \vec{\nabla}^2.$$

As it is the dot product of two 4-vectors, the d'Alembertian is a Lorentz invariant scalar.

Occasionally, in analogy with the 3-dimensional notation, the symbols $$\Box$$ and $$\Box^2$$ are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol $$\Box$$ is reserved for the d'Alembertian.

Some examples of the 4-gradient as used in the d'Alembertian follow:

In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): $$\left[(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = \left[\left(\frac{\partial_t^2}{c^2} - \vec{\nabla}^2\right) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0 $$

In the wave equation for the electromagnetic field (using Lorenz gauge $$ (\boldsymbol{\partial} \cdot \mathbf{A}) = \left(\partial_\mu A^\mu\right) = 0 $$): where:
 * In vacuum: $$ (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = \mathbf{0} = 0^{\alpha} $$
 * With a 4-current source, not including the effects of spin: $$(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = \mu_0 \mathbf{J} = \mu_0 J^{\alpha}$$
 * With quantum electrodynamics source, including effects of spin: $$(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = e\bar{\psi} \gamma^{\alpha} \psi$$
 * Electromagnetic 4-potential $$\mathbf{A} = A^{\alpha} = \left(\frac{\phi}{c}, \mathbf{\vec{a}}\right)$$ is an electromagnetic vector potential
 * 4-current density $$\mathbf{J} = J^{\alpha} = \left(\rho c, \mathbf{\vec{j}}\right)$$ is an electromagnetic current density
 * Dirac Gamma matrices $$\gamma^\alpha = \left(\gamma^0, \gamma^1, \gamma^2, \gamma^3\right) $$ provide the effects of spin

In the wave equation of a gravitational wave (using a similar Lorenz gauge $$\left(\partial_\mu h^{\mu\nu}_{TT}\right) = 0$$) $$(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) h^{\mu\nu}_{TT} = 0$$ where $$h^{\mu\nu}_{TT}$$ is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).

Further conditions on $$h^{\mu\nu}_{TT}$$ are:
 * Purely spatial: $$\mathbf{U} \cdot h^{\mu\nu}_{TT} = h^{0\nu}_{TT} = 0$$
 * Traceless: $$\eta_{\mu\nu} h^{\mu\nu}_{TT} = h^{\nu}_{TT\nu} = 0$$
 * Transverse: $$\boldsymbol{\partial} \cdot h^{\mu\nu}_{TT} = \partial_\mu h^{\mu\nu}_{TT} = 0$$

In the 4-dimensional version of Green's function: $$(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) G\left[\mathbf{X} - \mathbf{X'}\right] = \delta^{(4)}\left[\mathbf{X} - \mathbf{X'}\right]$$ where the 4D Delta function is: $$\delta^{(4)}[\mathbf{X}] = \frac{1}{(2 \pi)^4} \int d^4 \mathbf{K} e^{-i(\mathbf{K} \cdot \mathbf{X})}$$

As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

$$\int_\Omega d^4X \left(\partial_\mu V^\mu\right) = \oint_{\partial \Omega} dS \left(V^\mu N_\mu\right)$$ or $$\int_\Omega d^4X \left(\boldsymbol{\partial} \cdot \mathbf{V}\right) = \oint_{\partial \Omega} dS \left(\mathbf{V} \cdot \mathbf{N}\right)$$ where
 * $$\mathbf{V} = V^\mu$$ is a 4-vector field defined in $$\Omega$$
 * $$\boldsymbol{\partial}\cdot\mathbf{V} = \partial_\mu V^\mu$$ is the 4-divergence of $$V$$
 * $$\mathbf{V}\cdot\mathbf{N} = V^\mu N_\mu$$ is the component of $$V$$ along direction $$N$$
 * $$\Omega$$ is a 4D simply connected region of Minkowski spacetime
 * $$\partial \Omega = S$$ is its 3D boundary with its own 3D volume element $$dS$$
 * $$\mathbf{N} = N^\mu$$ is the outward-pointing normal
 * $$d^4X = (c\,dt) \left(d^3x\right) = (c\,dt) (dx\,dy\,dz)$$ is the 4D differential volume element

As a component of the SR Hamilton–Jacobi equation in relativistic analytic mechanics
The Hamilton–Jacobi equation (HJE) is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle

The generalized relativistic momentum $$\mathbf{P_T}$$ of a particle can be written as $$\mathbf{P_T} = \mathbf{P} + q\mathbf{A}$$ where $$\mathbf{P} = \left(\frac{E}{c}, \vec{\mathbf{p}}\right)$$ and $$\mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right)$$

This is essentially the 4-total momentum $$\mathbf{P_T} = \left(\frac{E_T}{c}, \vec{\mathbf{p_T}}\right)$$ of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle $$\mathbf{P}$$, plus momentum due to interaction with the EM 4-vector potential $$\mathbf{A}$$ via the particle charge $$q$$.

The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action $$S$$. $$\mathbf{P_T} = -\boldsymbol{\partial} [S] = \left(\frac{E_T}{c}, \vec{\mathbf{p_T}}\right) = \left(\frac{H}{c}, \vec{\mathbf{p_T}}\right) = -\boldsymbol{\partial} [S] = -\left(\frac{\partial_t}{c}, -\vec{\boldsymbol{\nabla}}\right)[S]$$

The temporal component gives: $$E_T = H = -\partial_t[S]$$

The spatial components give: $$\vec{\mathbf{p_T}} = \vec{\boldsymbol{\nabla}}[S]$$

where $$H$$ is the Hamiltonian.

This is actually related to the 4-wavevector being equal the negative 4-gradient of the phase from above. $$K^\mu = \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = -\boldsymbol{\partial} [\Phi]$$

To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum: $$\mathbf{P} \cdot \mathbf{P} = (m_0 c)^2$$

But from the minimal coupling rule: $$\mathbf{P} = \mathbf{P_T} - q\mathbf{A}$$

So: $$\begin{align} \left(\mathbf{P_T} - q\mathbf{A}\right) \cdot \left(\mathbf{P_T} - q\mathbf{A}\right) = \left(\mathbf{P_T} - q\mathbf{A}\right)^2 &= \left(m_0 c\right)^2 \\ \Rightarrow \left(-\boldsymbol{\partial}[S] - q\mathbf{A}\right)^2 &= \left(m_0 c\right)^2 \end{align}$$

Breaking into the temporal and spatial components: $$\begin{align} && \left(-\frac{\partial_t[S]}{c} - \frac{q \phi}{c}\right)^2 - (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 &= (m_0 c)^2 \\ &\Rightarrow & (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 - \frac{1}{c^2}(-\partial_t[S] - q \phi)^2 + (m_0 c)^2 &= 0 \\ &\Rightarrow & (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 - \frac{1}{c^2}(\partial_t[S] + q \phi)^2 + (m_0 c)^2 &= 0 \end{align}$$

where the final is the relativistic Hamilton–Jacobi equation.

As a component of the Schrödinger relations in quantum mechanics
The 4-gradient is connected with quantum mechanics.

The relation between the 4-momentum $$\mathbf{P}$$ and the 4-gradient $$\boldsymbol{\partial}$$ gives the Schrödinger QM relations. $$\mathbf{P} = \left(\frac{E}{c}, \vec{p}\right) = i\hbar \boldsymbol{\partial} = i\hbar \left(\frac{\partial_t}{c}, -\vec{\nabla}\right)$$

The temporal component gives: $$E = i\hbar \partial_t$$

The spatial components give: $$\vec{p} = -i\hbar \vec{\nabla}$$

This can actually be composed of two separate steps.

First:

$$\mathbf{P} = \left(\frac{E}{c},\vec{p}\right) = \hbar \mathbf{K} = \hbar \left(\frac{\omega}{c},\vec{k}\right)$$ which is the full 4-vector version of:

The (temporal component) Planck–Einstein relation $$E = \hbar \omega$$

The (spatial components) de Broglie matter wave relation $$\vec{p} = \hbar \vec{k}$$

Second:

$$\mathbf{K} = \left(\frac{\omega}{c},\vec{k}\right) = i \boldsymbol{\partial} = i \left(\frac{\partial_t}{c}, -\vec{\nabla}\right)$$ which is just the 4-gradient version of the wave equation for complex-valued plane waves

The temporal component gives: $$\omega = i \partial_t$$

The spatial components give: $$\vec{k} = - i \vec{\nabla}$$

As a component of the covariant form of the quantum commutation relation
In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).

 * According to: $$\left[P^\mu, X^\nu\right] = i \hbar \left[\partial^\mu, X^\nu\right] = i \hbar \partial^\mu\left[X^\nu\right] = i \hbar \eta^{\mu \nu}$$
 * Taking the spatial components, $$\left[p^j, x^k\right] = i \hbar \eta^{j k}$$
 * Since $$\eta^{\mu \nu} = \operatorname{diag}[1,-1,-1,-1]$$, $$\left[p^j, x^k\right] = - i \hbar \delta^{j k}$$
 * Since $$[a, b] = -[b, a]$$, $$\left[x^k, p^j\right] = i \hbar \delta^{k j}$$
 * And, relabeling indices gives the usual quantum commutation rules: $$\left[x^j, p^k\right] = i \hbar \delta^{j k}$$

As a component of the wave equations and probability currents in relativistic quantum mechanics
The 4-gradient is a component in several of the relativistic wave equations:

In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): $$\left[\left(\partial^\mu \partial_\mu\right) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0$$

In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons): $$\left[i \gamma^\mu \partial_\mu - \frac{m_0 c}{\hbar}\right] \psi = 0 $$

where $$\gamma^\mu$$ are the Dirac gamma matrices and $$\psi$$ is a relativistic wave function.

$$\psi$$ is Lorentz scalar for the Klein–Gordon equation, and a spinor for the Dirac equation.

It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric: $$\left\{\gamma^\mu, \gamma^\nu\right\} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu}I_4 $$

Conservation of 4-probability current density follows from the continuity equation: $$\boldsymbol{\partial} \cdot \mathbf{J} = \partial_t \rho + \vec{\boldsymbol{\nabla}} \cdot \vec{\mathbf{j}} = 0$$

The 4-probability current density has the relativistically covariant expression: $$J_\text{prob}^\mu = \frac{i\hbar}{2m_0}\left(\psi^* \partial^\mu\psi - \psi\partial^\mu \psi^*\right)$$

The 4-charge current density is just the charge ($q$) times the 4-probability current density: $$J_\text{charge}^\mu = \frac{i\hbar q}{2m_0}\left(\psi^* \partial^\mu\psi - \psi\partial^\mu\psi^*\right)$$

As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity
Relativistic wave equations use 4-vectors in order to be covariant.

Start with the standard SR 4-vectors:
 * 4-position $$\mathbf{X} = \left(ct, \vec{\mathbf{x}}\right)$$
 * 4-velocity $$\mathbf{U} = \gamma\left(c, \vec{\mathbf{u}}\right)$$
 * 4-momentum $$\mathbf{P} = \left(\frac{E}{c}, \vec{\mathbf{p}}\right)$$
 * 4-wavevector $$\mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right)$$
 * 4-gradient $$\boldsymbol{\partial} = \left(\frac{\partial_t}{c}, -\vec{\boldsymbol{\nabla}}\right)$$

Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar:
 * 4-velocity $$\mathbf{U} = \frac{d}{d\tau} \mathbf{X}$$, where $$\tau$$ is the proper time
 * 4-momentum $$\mathbf{P} = m_0 \mathbf{U}$$, where $$m_0$$ is the rest mass
 * 4-wavevector $$\mathbf{K} = \frac{1}{\hbar} \mathbf{P}$$, which is the 4-vector version of the Planck–Einstein relation & the de Broglie matter wave relation
 * 4-gradient $$\boldsymbol{\partial} = -i \mathbf{K}$$, which is the 4-gradient version of complex-valued plane waves

Now, just apply the standard Lorentz scalar product rule to each one: $$\begin{align} \mathbf{U} \cdot \mathbf{U} &= c^2 \\ \mathbf{P} \cdot \mathbf{P} &= (m_0 c)^2 \\ \mathbf{K} \cdot \mathbf{K} &= \left(\frac{m_0 c}{\hbar}\right)^2 \\ \boldsymbol{\partial} \cdot \boldsymbol{\partial} &= \left(\frac{-i m_0 c}{\hbar}\right)^2 = -\left(\frac{m_0 c}{\hbar}\right)^2 \end{align}$$

The last equation (with the 4-gradient scalar product) is a fundamental quantum relation.

When applied to a Lorentz scalar field $$\psi$$, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations: $$\left[\boldsymbol{\partial} \cdot \boldsymbol{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0$$

The Schrödinger equation is the low-velocity limiting case ($|v| ≪ c$) of the Klein–Gordon equation.

If the quantum relation is applied to a 4-vector field $$A^\mu$$ instead of a Lorentz scalar field $$\psi$$, then one gets the Proca equation: $$\left[\boldsymbol{\partial} \cdot \boldsymbol{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]A^\mu = 0^\mu$$

If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation: $$[\boldsymbol{\partial} \cdot \boldsymbol{\partial}]A^\mu = 0^\mu$$

More complicated forms and interactions can be derived by using the minimal coupling rule:

As a component of the RQM covariant derivative (internal particle spaces)
In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist.

The version known from classical EM (in natural units) is: $$D^\mu = \partial^\mu - i g A^\mu$$

The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is:

$$D^\mu = \partial^\mu - i g_1 \frac{1}{2} Y B^\mu - i g_2 \frac{1}{2}\tau_i \cdot W_i^\mu - i g_3 \frac{1}{2} \lambda_a \cdot G_a^\mu$$ or $$\mathbf{D} = \boldsymbol{\partial} - i g_1 \frac{1}{2} Y \mathbf{B} - i g_2 \frac{1}{2} \boldsymbol{\tau}_i \cdot \mathbf{W}_i - i g_3 \frac{1}{2} \boldsymbol{\lambda}_a \cdot \mathbf{G}_a$$

where the scalar product summations ($$\cdot$$) here refer to the internal spaces, not the tensor indices:
 * $$B^\mu$$ corresponds to U(1) invariance = (1) EM force gauge boson
 * $$W_i^\mu$$ corresponds to SU(2) invariance = (3) weak force gauge bosons (i = 1, …, 3)
 * $$G_a^\mu$$ corresponds to SU(3) invariance = (8) color force gauge bosons (a = 1, …, 8)

The coupling constants $$(g_1, g_2, g_3)$$ are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the $$g_i$$ are fixed for one representation, they are known for all representations.

These internal particle spaces have been discovered empirically.

Derivation
In three dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to 4 dimensions should be: $$\partial^\alpha \overset{?}{=} \left( \frac{\partial}{\partial t}, \vec{\nabla} \right),$$ which is incorrect.

However, a line integral involves the application of the vector dot product, and when this is extended to 4-dimensional spacetime, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of spacetime. In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention $$\eta^{\mu\nu} = \operatorname{diag}[1,-1,-1,-1]$$). The factor of (1/c) is to keep the correct unit dimensionality, [length]$−1$, for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient: $$\partial^\alpha = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right)$$

Note about References
Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use $$m$$ for invariant rest mass, others use $$m_0$$ for invariant rest mass and use $$m$$ for relativistic mass. Many authors set factors of $$c$$ and $$\hbar$$ and $$G$$ to dimensionless unity. Others show some or all the constants. Some authors use $$v$$ for velocity, others use $$u$$. Some use $$K$$ as a 4-wavevector (to pick an arbitrary example). Others use $$k$$ or $$\mathbf{K}$$ or $$k^\mu$$ or $$k_\mu$$ or $$K^\nu$$ or $$N$$, etc. Some write the 4-wavevector as $$\left(\frac{\omega}{c}, \mathbf{k}\right)$$, some as $$\left(\mathbf{k}, \frac{\omega}{c}\right)$$ or $$\left(k^0, \mathbf{k}\right)$$ or $$\left(k^0, k^1, k^2, k^3\right)$$ or $$\left(k^1, k^2, k^3, k^4\right)$$ or $$\left(k_t, k_x, k_y, k_z\right)$$ or $$\left(k^1, k^2, k^3, i k^4\right)$$. Some will make sure that the dimensional units match across the 4-vector, others do not. Some refer to the temporal component in the 4-vector name, others refer to the spatial component in the 4-vector name. Some mix it throughout the book, sometimes using one then later on the other. Some use the metric (+ − − −), others use the metric (− + + +). Some don't use 4-vectors, but do everything as the old style E and 3-space vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style throughout the whole derivation.

title: Four-gradient


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Zhongguancun

Zhongguancun (Chinese: 中关村) is a major technology hub in the Haidian District, Beijing, China.

Zhongguancun occupies a band between the northwestern Third Ring Road and the northwestern Fourth Ring Road in the northwestern part of Beijing. Zhongguancun is sometimes known as China's Silicon Valley. The place is also the center of the Beijing-Tianjin-Shijiazhuang Hi-Tech Industrial Belt.

History

Chen Chunxian envisioned Zhongguancun, which then became a well-known technology hub 30 years later. Chunxian, a member of the Chinese Academy of Sciences (CAS), conceived of a Silicon Valley in China following a government-sponsored trip to Boston and Silicon Valley, United States.

Zhongguancun became known as "Electronics Avenue" in the early 1980s, due to its information technology markets along a central, crowded street.

Zhongguancun was recognized by the central government of China in 1988, and officially named "Beijing High-Technology Industry Development Experimental Zone". In 1999, Zhongguancun became the "Zhongguancun Science & Technology Zone" with seven parks: Haidian, Fengtai, Changping, Electronics City (in Chaoyang), Yizhuang, Desheng, and Jianxiang. The original Zhongguancun became known as the Haidian Park of the Zhongguancun Zone.

Notable education and research centers
Zhongguancun has an association with China's two most prestigious universities, Peking University and Tsinghua University, along with Beijing Normal University and the Chinese Academy of Sciences, all of which are in close proximity.

Secondary schools in Zhongguancun include Affiliated High School of Peking University and High School Affiliated to Renmin University of China.

Government infrastructure
The Zhongguancun Administrative Committee oversees the city.

The State Administration of Foreign Experts Affairs (SAFEA) has its headquarters in Zhongguancun.

Notable companies and landmarks
Hailong Market, Guigu Market, Taipingyang Market, Dinghao Market and Kemao Market are prominent IT and electronics technology bazaars, noted for "shops with a shop", where bargaining is the norm. According to the 2004 Beijing Statistical Yearbook, there were over 12,000 high-tech enterprises operating in Zhongguancun's seven parks, with 489,000 technicians employed.

The most notable companies that grew up in Zhongguancun are Stone Group, Founder Group, Glodon and Lenovo Group, each founded from 1984 to 1985. Stone was the first successful technology company to be operated by private individuals outside of the government of China. Founder is a technology company that spun off Peking University. Lenovo Group spun off from Chinese Academy of Sciences with Liu Chuanzhi, a hero of Zhongguancun and current chairman, eventually taking the helm. Lenovo purchased IBM's PC division for $1.75 billion in 2005, making it the world's third-largest PC maker. Both Founder and Lenovo Group maintain strong connections to their academic backers, who are significant shareholders.

Many world-renowned technology companies built their Chinese headquarters and research centers in Zhongguancun Technology Park, such as Google, Intel, AMD, Oracle Corporation, Motorola, Cogobuy Group, IBM, MySpace, Sony, Solstice, Glodon and Ericsson. Microsoft has built its Chinese research headquarters in the park that costs $280 million and can accommodate 5000 employees, which was completed in April 2011, and now houses Microsoft Research Asia.

The development center of Loongson, which is China's first general-purpose microprocessor design, is also in the Zhongguancun area.

Everbright International has its Beijing office in the Beijing International Building (北京国际大厦) in Zhongguancun.

Many conferences are held in this location, including the annual ChinICT conference - which is the largest Information technology Development and Entrepreneurship event in China.

A frequent tourist destination is the Haidian Christian Church, designed by Hamburg-based architects Gerkan, Marg and Partners.

Transportation
Beijing Subway Line 4 runs through the Zhongguancun area with stops at Zhongguancun Station and Haidianhuangzhuang Station. Haidianhuangzhuang is also a transfer station with Line 10. In addition, Zhongguancun is served by many of Beijing's buses.

title: Zhongguancun


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