What is Einstein equation: Definition and 28 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. J

    I Schwarzschild Geometry: Einstein Tensor & Mass Density

    The Einstein tensors for the Schwarzschild Geometry equal zero. Why do they not equal something that has to do with the central mass, given that the Einstein equations are of the form: Curvature Measure = Measure of Energy/Matter Density?
  2. Dario56

    Diffusion, Migration and Einstein Equation

    In the textbook: Electrochemical Systems by Newman and Alyea, 3rd edition, chapter 11.9: Moderately Dilute Solutions, equation for the mole flux of the component ##i## is given by: $$ N_i = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ + c_i v \tag {1}$$ where ##u_i## is the ionic mobility...
  3. T

    I Is the sign of the Einstein equation right in Chandrasekhar's Black holes?

    In Chandrasekhar's book, The Mathematical Theory of Black Holes. The sign of Einstein equations is minus "-" , Eq. (1-236). However, the sign of Riemann and Ricci tensor are the same as MTW's book. The sign of Einstein equations in MTW's book are "+"! Is there a error?
  4. C

    Proving ##C## is constant in 4-dim ##R_{\mu\nu}=Cg_{\mu\nu}##

    This question wasn't particularly hard, so I assume metric compatibility and input Ricci tensor to the left side of Einstein's equation. $$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=Cg_{\mu\nu}-\frac{1}{2} (4C)g_{\mu\nu}=-Cg_{\mu\nu}$$ Then apply covariant derivative on both side...
  5. E

    I Left-right arrow in Einstein equation

    Short, probably stupid, question; what does this left-right arrow ##a \leftrightarrow b## at the bottom-right mean? [It's this paper for ref.] Does it just mean the last term repeated, except this time with ##a## and ##b## indices exchanged? Thanks.
  6. T

    I Solving the Spherically Symmetric Einstein Equation

    Can be Einstein equation rewrited into some simpler form, when suppose only spherically symmetric (but not necessarily stationary) distribution of mass-energy ? If yes, is there some source to learn more about it ? Thank you. edit: by simpler form I mean something with rather expressed...
  7. sush

    I Why E=mc^2 is different from E=1/2 mv^2?

    Why kinetic energy is ½ m v2? Why it is different from Einstein’s equation for Energy E= m c2?
  8. C

    I Deductions of Formulas for Energy

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  9. D

    I Newtonian limit of Schwarzschild metric

    If I am asked to show that the tt-component of the Einstein equation for the static metric ##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?
  10. G

    Problem about equality of energy formulas

    Above they use the equality that the massless energy hv is equal to einsteins energy. Both have the same velocity. How can not this mean that einsteins formula has more energy then the other. I have added a proof of einsteins formula just in case it can be used: In the proof above I don't...
  11. V

    Breakdown of Correspondence Principle: Null Dust Case

    In both quantum and general relativity theories we are used to provide results in the "limited" conditions to demonstrate a correspondence between new and old formalism. For instance deflection of light of a star due to Sun in GR is double the amount given in classical theory. Yet I have...
  12. loops496

    Scalar Field Solution to Einsteins Equations

    Homework Statement Compute $$T_{\mu\nu} T^{\mu\nu} - \frac{T^2}{4}$$ For a massless scalar field and then specify the computation to a spherically symmetric static metric $$ds^2=-f(r)dt^2 + f^{-1}(r)dr^2 + r^2 d\Omega^2$$Homework Equations $$4R_{\mu\nu} R^{\mu\nu} - R^2 = 16\pi^2 \left(...
  13. mileymo

    Gravity, Time Dilation & Black Holes: Q&A

    I'm wondering if a ship orbited a black hole for a year, how large would the black hole be for it to equal 10,000 years on earth?
  14. S.Daedalus

    Einstein Equation of State vs Gravity from Entanglement

    There's a somehow related set of issues I find myself pondering time and again: In 1995, Ted Jacobson derived Einstein's equations from thermodynamics across a horizon. Roughly, he showed that if the horizon's entropy is given by the Bekenstein-Hawking formula, then the second law of...
  15. binbagsss

    Solving Friedmann Equations with Cosmological Constant

    Hi, I'm looking at 'Lecture Notes on General Relativity' by Sean M.Carroll. I have a question about p. 227, solving for ##a(t)## in the dark energy case. So for dust and radiation cases it was Friedmann equations you solve. But in the case of a non-zero cosmological constant Eienstien equation...
  16. Baluncore

    Can Einstein's Field Equations Explain the Sinking Locomotive in Bolivian Sand?

    Do the Einstein field equations explain why this locomotive is sinking into the Bolivian sand?
  17. binbagsss

    Cosmological Constant, Einstein equation Quick Question

    So Einstein Equation: ##G_{uv}= 8 \pi G T_{uv} ##, Justifying the cosmological constant can be included is done by noting that ## \bigtriangledown^{a}g_{ab} =0 ## and so including it on the LHS, conservation of energy-momentum tensor still holds. I'm not sure why ## \bigtriangledown^{a}g_{ab}...
  18. binbagsss

    Einstein Equation Quick sign question

    Some sources seem to have: ##G_{uv}=8\pi G T_{uv} ## Whereas others have: ##G_{uv}=-8\pi G T_{uv} ## I thought that it may have been covered by how ##G_{uv}## is defined on the sources, but in both cases it is given as ## G_{uv}=R_{uv} - \frac{1}{2}g_{uv}R ## I'm confused. Thanks.
  19. L

    Calculating Einstein Equation from Lagrangian Eqn

    I have a 3d system with Lagrangian e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q) From this I want to calculate the Einstein equation by performing the Euler-Lagrange procedure. First of all, I move the 3d dreibein to the RHS and then I...
  20. M

    Is the Einstein equation E=Mc^2 correct?

    I wonder if Einstein equation E=mc^2 was used when the A bombes were designed and tested. How was the output estimated? Was all the matter thransformed into energy, as the equation indicates? Were there any restrictions to how this equation ( or nny other) was used? Thanks, Michael
  21. M

    Motivation behind vacuum Einstein equation

    Hi, I am using Hartle to study GR and at one point, there is a leap that I don't understand. He finds the result for the geodesic deviation equation and introduces the Riemann curvature tensor. Then, we are told that there is an object called the Ricci curvature tensor which is a...
  22. P

    What does the mass represent in the Schwarzschild metric for vacuum solutions?

    Sorry if I am asking a too trivial question! I am having a confusion regarding the following-The solution to Einstein equation in vacuum is given by the Schwarzschild metric. However, what does the mass represent in the metric in Schwarzschild coordinate? Whose mass is it and how does it enter...
  23. R

    Einstein equation and particle jets

    Does anyone know if Einstein equation predicts the existence of particle jets fired by massive black holes observed by NASA from active galaxies? If yes, what would be the predicted relation between the black hole mass and rotation speed to the flux of matter ejected, how long these jets...
  24. L

    Solving Einstein Equation Confusion: My Notes vs Freidmann Equations

    My notes for a particular course say that G_{ab} - \Lambda g_{ab}= x T_{ab} where x=\frac{8 \pi G}{c^4} Then they say that the trace of this is -R+4 \Lambda=xT What? Surely that's only possible if we have +\Lambda g_{ab} as I have seen in every other text I've ever read? However, when we...
  25. E

    Einstein equation point mass solution

    I'm just getting a taste of computational GR, and I have a question regarding the metric for the single point mass solution for the einstein equation. The metric in spherical coordinates for a point mass at r=0 is \eta=\left(\begin{matrix}-\left(1-\frac{2GM}{c^2r}\right) & 0 & 0 & 0 \\ 0 &...
  26. M

    Thermodynamics of Spacetime: The Einstein Equation of State?

    Hi, I have recently been delving into Quantum gravity related material... and I came across a paper by Ted Jacobson "Thermodynamics of Spacetime: The Einstein Equation of State" http://arxiv.org/abs/gr-qc/9504004 and as far as I understand the argument it is very impressive who would have...
  27. G

    Using the Einstein Equation to Solve Problems in GR

    Hi, i have read a introduction book to general relativity and i want to ask a question: how do physicists use the Einstein equation to solve problems? I mean how can we, starting with a given energy-momentum tensor, find the metric? Thanks!
  28. V

    Finding a Kerr Metric Using the Einstein Equation

    How can I obtain a Kerr metric by using the Einstein equation?
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