Talk:Einstein field equations

Whether the geodesic equation is dependent on EFE?
To : You said "GE is independent of EFE." and provided a reference. Although many people say that they are independent, none-the-less they are not. See Post-Newtonian expansion which has an external link to "ON THE MOTION OF PARTICLES IN GENERAL RELATIVITY THEORY" by Albert Einstein and Leopold Infeld. Basically, each elementary particle is viewed as a (possibly charged) (possibly rotating) black-hole &mdash; a gravitational monopole. The ability to match these local solutions to the surrounding metric field, without introducing gravitational dipoles and thus negative energy, uniquely determines the motion of the particles. JRSpriggs (talk) 21:02, 27 September 2011 (UTC)
 * That shows that disturbances in the metric are determined by the EFEs, as we would expect, but says nothing about actual particles. I think most physicists would concede that particles are more than just metric distortions, despite Einstein's love of the idea. -- cheers, Michael C. Price talk 21:47, 27 September 2011 (UTC)
 * In my understanding work like A Rigorous Derivation of Gravitational Self-force (Gralla, Wald) classically settles this in favour of dependence. Virtual Neutrino (talk) 16:14, 15 November 2019 (UTC)

Please note the image,

http://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Spacetime_curvature.png/250px-Spacetime_curvature.png

Is inacurate, as the gridding should converge towards the core of the depicted Mass.. — Preceding unsigned comment added by 120.148.18.177 (talk) 00:28, 17 July 2012 (UTC)

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(im-)proper noun?
The term Einstein field equations (in the title and in the article) should be capitalized, as "Einstein Field Equations", if it is to be considered a name. Otherwise it should be written as "Einstein's field equations". Either way, as it is it does not seem correct. Could somebody fix this, or explain why I am wrong? Aleck, Smart (talk) 14:23, 23 May 2019 (UTC)
 * I don't think there's anything wrong with "Einstein field equations". Compare with "Lorentz transformation". Check the litrerature:


 * {| class="wikitable" style="text-align: center"

! Google !! Scholar !! Books
 * "Einstein field equations"
 * 
 * 
 * "Lorentz transformation"
 * 
 * 
 * }
 * ... with or without the 's, and sufficiently in lower case. - DVdm (talk) 14:47, 23 May 2019 (UTC)
 * ... and Bohr radius and Planck mass and ... - DVdm (talk) 14:50, 23 May 2019 (UTC)
 * I agree with DVdm. In this form it is preceded by the definite article ("the Einstein field equations"), and I don't see what might feel incorrect with this.  The form of standard descriptive phrases like this without capitalization and without the possessive form seems to be the modern norm for "named" concepts in physics.  —Quondum 16:31, 20 August 2020 (UTC)
 * Yes, there's "the Schrödinger equation" and "the Dirac equation" and "the Klein–Gordon equation", for example. The one familiar exception I can think of is "Maxwell's equations", where the possessive form is common, although the version with the definite article also occurs, and sometimes the two coexist in the same passage. XOR&#39;easter (talk) 19:16, 20 August 2020 (UTC)
 * ... and Bohr radius and Planck mass and ... - DVdm (talk) 14:50, 23 May 2019 (UTC)
 * I agree with DVdm. In this form it is preceded by the definite article ("the Einstein field equations"), and I don't see what might feel incorrect with this.  The form of standard descriptive phrases like this without capitalization and without the possessive form seems to be the modern norm for "named" concepts in physics.  —Quondum 16:31, 20 August 2020 (UTC)
 * Yes, there's "the Schrödinger equation" and "the Dirac equation" and "the Klein–Gordon equation", for example. The one familiar exception I can think of is "Maxwell's equations", where the possessive form is common, although the version with the definite article also occurs, and sometimes the two coexist in the same passage. XOR&#39;easter (talk) 19:16, 20 August 2020 (UTC)

Inference as stated is incomplete
"The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy." Strictly speaking, this is invalid, or at least the argument given is incomplete: one cannot draw the conclusion of equivalence from only one component of the stress–energy tensor. One needs to work from some constraint on the form that the vacuum energy stress–energy tensor takes to reach the conclusion given, for example that this tensor is the same in every frame of reference, thus allowing one to deduce the other components of the tensor. I do not have a reference to hand to complete the stated logic. —Quondum 16:47, 20 August 2020 (UTC)


 * Vacuum is understood to be the state of maximum symmetry. This is only possible if
 * $$T_{\mu \nu}^\mathrm{(vac)} \propto g_{\mu \nu} \,.$$
 * OK? JRSpriggs (talk) 19:13, 20 August 2020 (UTC)


 * This is not obvious to the reader of this article; indeed, "vacuum energy" is undefined in this article: a reader should not need to follow a link and read another article to determine this key fact. So a statement to the effect of what you have just said would be needed in this article.  —Quondum 22:19, 20 August 2020 (UTC)

✅ —Quondum 20:43, 24 August 2020 (UTC)

"fix units of pressure"
seems to suggest that "energy density" does not have the same units as "pressure". This kind of thing is easy for me to get wrong (kinda fiddly), but I am failing to make sense of this. The way I figure it, a quantity called "energy density" has the SI unit J⋅m$−3$, and a quantity called "pressure" has the SI unit N⋅m$−2$ = J⋅m$−3$, which is the same unit. I am aware that the components of the tensor corresponding to these quantities might have other units by convention, but here we are talking about SI quantities, not tensor components. I'd be glad to have an error on my part pointed out, but for the moment I am just not getting it. The way I see it, the units of $Λ$ (either m$–2$ or s$–2$) affects the exponent of $c$ on the very right, but we will always have $ρvac = –pvac$ without any other constant factors. —Quondum 20:09, 22 August 2020 (UTC)


 * Sorry, I did not notice "energy density". I assumed that &rho; referred to density of relativitistic mass as it does at stress-energy tensor. JRSpriggs (talk) 22:00, 23 August 2020 (UTC)


 * I guess the article is missing something to indicate which convention of units is in use for each tensor. There are enough variations to make my head spin.  —Quondum 22:10, 23 August 2020 (UTC)

With the adjustment from $κ = 8πG/c4$ to $κ = 8πG/c2$ (which, to be sure, I like since it is the same as Einstein's original choice), the footnote [6], which is intended to refer to the other common definition, is now out of sync. Since I am not familiar with how the units of the tensor components are chosen in common use, I cannot update this with any confidence. Could someone take a look at this? —Quondum 00:58, 24 August 2020 (UTC)


 * ✅ (by JRSpriggs)  —Quondum 20:41, 24 August 2020 (UTC)

6 equations?
The article says there are 6 equations, but nowhere in the article does it identify which are the 6 equations? Voproshatel (talk) 18:48, 10 December 2020 (UTC)


 * Where does the article say that? Describe the section/paragraph/sentence.
 * See the section Talk:Einstein field equations above. JRSpriggs (talk) 02:10, 11 December 2020 (UTC)


 * Note that the article says "The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6." - DVdm (talk) 14:40, 11 December 2020 (UTC)


 * See Riemann curvature tensor. If you take the covariant divergence of the Einstein field equations, the right-hand side becomes zero by virtue of the conservation of energy and momentum. The Bianchi identities make the left-hand side reduce to zero also, giving four identities.
 * It has been so long since I saw this that I forgot the details, I will have to look it up again. JRSpriggs (talk) 16:24, 11 December 2020 (UTC)
 * Welcome to the club - DVdm (talk) 16:40, 11 December 2020 (UTC)

Leading sentence is misleadingly oversimplified
I find the leading sentence grates - "In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.[1]". The equations do _not_ do this. They relate the geometry of spacetime to the distribution and motion of energy-momentum. I suggest that some way is found to make the lead correct while perhaps not scaring people off with terms like "stress-energy tensor" so early. Elroch (talk) 14:25, 10 November 2021 (UTC)

Article issues and classification

 * The article is tagged as having "unsourced statements from October 2014" and "too technical from May 2021". The criteria #1 states; The article is suitably referenced, with inline citations. It has reliable sources, and any important or controversial material which is likely to be challenged is cited. There is far too much unsourced content. Criteria #4 states, "The article is reasonably well-written.". The article needs to be reassessed.

Citation of Einstein's paper after "The Einstein field equations (EFE) may be written in the form:"
The quoted paper by Einstein is from 1916, but the equation includes the cosmological constant, which Einstein only introduced in his 1917 paper (https://articles.adsabs.harvard.edu/pdf/1917SPAW.......142E). I do not know how referencing works on wikipedia, so I do not want to add the source myself. 145.107.121.33 (talk) 13:04, 27 June 2023 (UTC)

Is the value for the gravitational constant too accurate?
In the section on mathematical form, the Einstein Gravitational constant is given a value accurate to 13 significant digits:

κ = 2.076647442844 × 10-43 N-1

This is a question (not a statement of certain fact), but to the best of my understanding Newton's gravitational constant G is known only to 5-6 significant digits: please see NIST's "Newtonian constant of gravitation" page:

https://physics.nist.gov/cgi-bin/cuu/Value?bg.

Or maybe I am not up to date? But if I got it right, it would be more correct to state:

κ ≈ 2.07665 × 10-43 N-1

If the number is correct as is - could you add a reference? ￼ Thank you. Giladpn (talk) 11:30, 1 November 2023 (UTC)


 * Is the error in the NIST value due to some deficiency in the Cavendish experiment data or to some problems with other data in their data-base? If the latter, I would not want to trim the value given. JRSpriggs (talk) 19:53, 1 November 2023 (UTC)
 * Thanks for the response, I appreciate what you are saying.
 * A reference I do have supports a lower value for the accuracy (Precision measurement of the Newtonian gravitational constant - PMC (nih.gov)). I think it pretty clearly states that the available accuracy exists in more than a dozen experiments over the last few decades. Please consider.
 * What I know - and this has been true for many years now - is that Newton's G is known to 5-6 digits because Gravity is much weaker than other forces, and in a terrestrial lab experiment this is the best that can be done.
 * Another reference sheds light on the latest science. Measurements from gravitational waves show that the speed of light and the speed of GW is the same - to 15 significant digits. I am guessing that this latter fact confused the original author who thought that G was also known to such accuracy, but so sorry...
 * So just asking what is the reference for the higher accuracy? It is unreferenced as it stands.
 * Thank you! Giladpn (talk) 11:41, 2 November 2023 (UTC)
 * All the reliable sources I can find point to G having a 5-6 digit certainty, while the speed of GW vs c makes sense to be accurate down to 15 digits, since measuring the presence of a GW is much easier than measuring its exact strength. Measuring the exact strength of gravity would be difficult to do on Earth because of interference from large objects like mountains and the Moon, so it makes sense that it only has 5-6 significant digits.
 * My personal opinion is that changing the value of Einstein’s gravitational constant as listed in the article to only include this many digits is a good idea - we don’t actually know that is has that value because we don’t know what G is exactly. OverzealousAutocorrect (talk) 18:01, 29 February 2024 (UTC)
 * By definition, $$\frac{G}\kappa=\frac{c^4}{8\pi}\textrm{N}.$$ The right-hand value is known with infinite precision:
 * $$\frac{c^4}{8\pi}=321397838761514638193268935404913.60432793451267227{\cdot}{\cdot}{\cdot}\text{m}^4\text{s}^{{-}4}$$
 * Therefore, the relative uncertainty in the values of $$G$$ and $$\kappa$$ is exactly equal. It is well-known and well-documented that $$G$$ is very hard to determine, which is expressed in the CODATA uncertainty. Just look at the table captioned "Recommended values for G" in . The high precision with which $$\kappa$$ is presented here, which is not reflected in its presentation in reliable sources,* is therefore totally unwarranted. --Lambiam 06:13, 5 March 2024 (UTC)
 * 
 * * As presented in reliable sources, we find, for example,,, and, in the cgs system, [sic].