What is Einstein field equation: Definition and 25 Discussions

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.The equations were first published by Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor).Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.
As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light.Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.

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  1. cianfa72

    I General Relativity and the precession of the perihelion of Mercury

    Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury. My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.
  2. Pyter

    I What's the underlying frame of the Einstein's Field Equation?

    Hello all, I have a question on a pivotal concept of GR that I've never managed to fully grasp. In what coordinate system is the Einstein's Field Equation set up and solved? I've always assumed it's an Euclidean 4D space, whose metric is irrelevant because we are dealing with scalar...
  3. D

    I Cartan's Understanding of Einstein Field Equation

    About a week ago I was reading about Cartan's geometric interpretation of the Einstein Field Equation Gij + Λgij = κTij According to Cartan, this equation expresses the idea (sum of moments of rotation for the faces of a little 3-cube) = 8π * (amount of energy-momentum within that 3-cube) As...
  4. Lilian Sa

    Star collapse in general relativity — pressure as a function of star radius

    What I've done is using the TOV equations and I what I found at the end is: ##e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)## so I am sure that this is not right, if someone can help me knowing it I really apricate it :)
  5. diazdaiz

    Please help me to be able to read the Christoffel symbol in EFE

    in video "Einstein Field Equation - for Beginner!" by "DrPhysicsA" on youtube, in 01:10:56, the christoffel symbol equation is written, then i see in "Physics Videos by Eugene Khutoryansky" video with title "Einstein's Field Equations of General Relativity Explained" in minute 05:02 on how the...
  6. Vick

    Coding Riemann (0,4) Equation in Python - Help Needed!

    Hi, I've coded Riemann tensor in python successfully. However, I recently stumbled onto another Riemann equation for the valence (0,4) as shown in the following link: Riemann (0,4) I'm having troubled coding the last part after the partial derivatives and the plus sign. Can anyone help me? Thanks
  7. V

    I Proving Effects of Stress-Energy Tensor on Curvature

    Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...
  8. D

    I How do I apply weak field approximation to the Einstein field equation?

    Actually from the Einstein field equation I am trying to compute the R00 component and get at a point R00=k(.5Tg00-T00) where,T=gαβTαβ now I am trying to apply the weak field approximation ,how will I proceed?
  9. T

    I How does parallel transportation relates to Rieman Manifold?

    Source: Basically the video talk about how moving from A to A'(which is basically A) in an anticlockwise manner will give a vector that is different from when the vector is originally in A in curved space. $$[(v_C-v_D)-(v_B-v_A)]$$ will equal zero in flat space...
  10. T

    How is the Einstein field equation converted to its tensor form?

    ∇2g00 = 8πGρ According to drphysics, as this is not a tensor equation, we need to change it such that it fits general relativity. Gμv = 8πGTμv T00 is the energy density. The conversion of the density(ρ) to T00 is it done through E=mc2? ρ=E/c2 And since is T00, it will be over c4 instead of...
  11. A

    What is the purpose of the Einstein stress-energy tensor?

    Hello I'm new here on this forum and on physics too. I have problem on Einstein famous equation I have a problem on the last component Tαβ I know that tensor name is Einstein stress-energy tensor and I know that Tαβ...
  12. T

    Einstein Field Equation: Taking into Account Special Relativity & Light Speed

    How does efe takes into special relativity? Is it due to efe been based on spacetime which takes into account the time component? But how does it takes into account the limiting speed of light?
  13. T

    Solve Einstein Field Equation without Vacuum Solution

    How to solve EFE without using the vacuum solution like how you did in schwarzschild metric?
  14. T

    Does the Einstein Field Equation Apply to Small Particles like Electrons?

    What happens if you apply EFE to a small particle like electron? Is it where efe breaks down and have to be replaced with quantum gravity? My apologies for such a dumb qns as I just started.
  15. T

    Einstein Field Equation: Understanding Curvature and Geodesic Equations

    Is EFE an equation to help us find the curvature cause by mass? Like the schwarzchild metric? In addition, is it a must to use polar coordinate for EFE to work since it contains dr? Can we used the ordinary Euclidean coordinate in minkoswki space for EFE? If we can't is it because there is no...
  16. N

    Constructing solutions to Einstein Field Equation

    The swartzchild metric, the kerr metric etc. where constructed with specific goals in mind, how does one go about doing that with goals in mind? Any help or comments appreciated. I don't expect a guide, I doubt one exists,
  17. T

    Understanding Einstein Field Equation & Metric Tensor

    Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...
  18. M

    Constructing the Einstein Field Equation

    This question is on the construction of the Einstein Field Equation. In my notes, it is said that >The most general form of the Ricci tensor R_{ab} is R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab} where R is the Ricci scalar. Why is this the most general form (involving up to the second derivative...
  19. C

    A question of Einstein field equation

    I got some trouble from this question: For a given metric: ds2 =t-2(dx2-dt2), derive the energy-momentum tensor which satisfies the Einstein equation: Rαβ- 1/2Rgαβ=8\piGTαβ. I got the Ricci scalar R=2, but Tαβ=0 for all α,β. Does this means a curved spacetime without any source(energy-momentum...
  20. B

    Euler Lagrange equation as Einstein Field Equation

    I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation. First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) : T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p &...
  21. L

    How Einstein field equation becomes the Poisson equation?

    I want to show that ∇2ϕ=ρ/2, which governs gravity in Newtonian physics? I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271]. Solution refer to optional exercise as follows: Let R^ be the (0, 4) –tensor field physically...
  22. P

    Einstein Field Equation Solution - How?

    Hi, I just wonder is there good reference to show how to solve Einstein's Field Equation? It seems that his equation can generate many possibilities but what techniques that require us to study and solve it? Thanks Alex
  23. K

    Using Einstein field equation

    Gµv + Λgµv = (8πG/c4)Tµv I have several questions. what is the µv? when we use it today do we use the cosmological constant even though the universe isn't static or does it mean something different than einstein orignally thought? what are we measuring when we use this if theyre all constants?
  24. Q

    Einstein Field Equation purely in partial derivatives wrt x,y,z,t?

    Has anyone evaluated the Einstein Field Equation purely in partial derivatives wrt x,y,z,t? What does it look like?