# Einstein field equation Definition and 7 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. ### 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...
2. ### 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 :)

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...
4. ### A Riemann Tensor

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
5. ### I Stress-energy tensor contribution to 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...
6. ### 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...
7. ### 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...