What is Einstein field equations: Definition and 50 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. grav-universe

    I Equation of state for Einstein field equations

    Years ago I posted a thread where I solved for the exterior vacuum metric for a static spherical mass using only a single one of the unknown functions, A, B, or D, where D = C r^2, since they are inter-related. A moderator here graciously supplied the EFE's as A, B, and D relate to the energy...
  2. Sciencemaster

    I Adapting Schwarzschild Metric for Nonzero Λ

    So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...
  3. P

    A Einstein Field Equations: Spherical Symmetry Solution

    [Moderator's Note: Thread spin off due to topic and level change.] For a spherically symmetric solution, if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, the I believe solving the EFE would require spherical symmetry of the...
  4. Sciencemaster

    I Piecewise Functions in the Einstein Field Equations

    Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having...
  5. cianfa72

    I Mercury Perihelion Precession: Analytic Derivation

    HI, I'm curios about the analytic derivation of Mercury perihelion precession starting from EFE - Einstein Field Equation (or simply just from Schwarzschild solution of the EFE). Can you advise me about some source or online material to learn it ? Thanks.
  6. K

    I Number of Solutions of Einstein Field Equations w/ Zero Pressure

    Is it true that the Einstein Field Equations have an infinite number of solutions when the pressure is zero?
  7. E

    A Derivation of Einstein Field Equations w/o Poisson or Least Action

    I would be grateful if some one would consider my following thought and indicate to me the likely mistakes, which I cannot do. Following the paper “Why the Riemann Curvature Tensor needs twenty independent components” by David Meldgin UC Davis 2011, I understand that with a coordinates...
  8. ohwilleke

    I Benefits & Disadvantages of Palatini Action for General Relativity

    Most often, general relativity is formulated in terms of Einstein's field equations: whose terms are familiar to readers in this forum. But, I understand (and feel free to correct me or qualify my statement if I am incorrect) that it is also possible to describe general relativity with an...
  9. D

    I Weak Gravitational Field: Solving Einstein Field Eqs

    What do I have to do if I want the EFE's to approximate a weak gravitational field, where for example, an inversely proportional to the cube ( ##1 / r^3## ) of the distance law between the masses applies?
  10. Cathr

    I Is there an analog to Einstein's field equations for 2D?

    I am not familiar with tensors and I would like to know if it's possible to understand GR without using them. I imagine we use them to describe four-dimentional space-time, because a regular vector or matrix wouldn't be enough. Is there an analog of Einstein's equations for a 2D space (plane)...
  11. MattRob

    I EFE: Stress-Energy & Spacetime Curvature

    So, rather than causality and time travel paradoxes and the like that are usually discussed about relativity, I'm curious about something else. On one side of the Einstein Field Equations is the Stress-Energy Tensor, along with some constant coefficients (G, c^-4, etc), which essentially...
  12. T

    I Einstein Field Equations: PDEs or ODEs? - Thomas

    This past semester, I just took an introductory course on G.R., which translates to a lot of differential geometry and then concluding with Schwarzschild's solution. We really didn't do any cosmology. However, one of the themes that kept creeping up again and again is that in 4-dimensions...
  13. Leonardo Machado

    A Non static and isotropic solution for Einstein Field Eq

    Hello dear friends, today's question is: In a non static and spherically simetric solution for Einstein field equation, will i get a non diagonal term on Ricci tensor ? A R[r][/t] term ? I'm getting it, but not sure if it is right. Thanks.
  14. E

    I Help with Derivation of Linearized Einstein Field Eqns

    Hi all - I am trying to follow a derivation of the above. At some point I need to find gαβ for gαβ = ηαβ + hαβ with |hαβ|<<1 I am stuck. The text says gαβ = ηαβ - hαβ but I cannot figure out why. Can anybody help?
  15. A

    I Ricci tensor for Schwarzschild metric

    Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...
  16. A

    I Exploring the Ricci Tensor: Einstein Field Equations

    Hello I've been have been done some research about Einstein Field Equations and I want to get great perspective of Ricci tensor so can somebody explain me what Ricci tensor does and what's the mathmatical value of Ricci tensor.
  17. redtree

    A Assumptions of the Schwarzschild solution to the EFE

    Given that no assumption is of a point energy is necessary to derive the vacuum (Schwarzschild) solution to the EFE, why is the solution assumed to apply to spacetime surrounding a point energy?
  18. I

    Einstein Field Equations, how many?

    Hello, can somebody please help me understanding the following. Action of general relativity consists of two terms: action of gravitation, dependent on metric tensor and its derivatives; action of matter, say one freely moving point mass particle, dependent on particle coordinates and metric...
  19. dylanreynolds1

    Computational General Relativity

    Hello all, first post. I have come here to get second opinions on the program I have written to compute the Einstein Tensor (the Riemann Tensor and Ricci Tensor). I enjoy looking for solutions to the Einstein Field Equations, however computing them by hand is not realistic. I decided to write a...
  20. J

    Spacetime Curvature: Which Tensor Gives Coordinates?

    In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...
  21. meyol99

    Understanding Einstein Field equations?

    Hello dear Physicists, I am very curious about understanding the math and the nature properties of the Einstein Field Equations.What I need to know is,what concrete mathematical operations I need to know and understand,and have experience with to understend this theory.I'm a quick learner and...
  22. U

    Energy-Momentum Tensor of Perfect Fluid

    Homework Statement I am given this metric: ##ds^2 = - c^2dt^2 + a(t)^2 \left( dx^2 + dy^2 + dz^2 \right)##. The non-vanishing christoffel symbols are ##\Gamma^t_{xx} = \Gamma^t_{yy} = \Gamma^t_{zz} = \frac{a a'}{c^2}## and ##\Gamma^x_{xt} = \Gamma^x_{tx} = \Gamma^y_{yt} = \Gamma^y_{ty} =...
  23. U

    Einstein Equations of this metric

    Homework Statement [/B] (a) Find the christoffel symbols (b) Find the einstein equations (c) Find A and B (d) Comment on this metric Homework Equations \Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha...
  24. T

    Understanding Einstein Field Equations & Levi Civita Symbol/Jacobian Determinant

    How do you represent einstein field equations with levi civita symbol or jacobian determinant? I saw a lot of work that involves this but I don't know how and why. Besides how is the jacobian determinant related to the levI civita symbol?
  25. U

    Einstein Tensor - Particle at rest?

    Homework Statement (a)Find Christoffel symbols (b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors (c) Find zeroth component of Einstein Tensor Homework EquationsThe Attempt at a Solution Part (a)[/B] Let lagrangian be: -c^2 \left( \frac{dt}{d\tau}\right)^2 +...
  26. ShayanJ

    Fermionic Fields in Einstein Field Equations | Explained

    In the Einstein-Hilbert action wikipedia page, the following paragraph is written: I thought for treating spin, we need to consider Einstein-Cartan theory! This is really surprising to me. Can anyone suggest a paper or book that explains this in some detail? Thanks
  27. 2

    Deriving the Schwarzchild radius?

    I'm a bit confused about the derivation of the Schwarzschild radius. I can do it quite easily using Newton's Law of gravitation, but this law is only an approximation, so I am wondering whether the result I obtain, r_{s}=\frac{2GM}{c^{2}}, is an approximation or not. It seems to me that it...
  28. C

    Derivations of Einstein field equations

    Hello Everyone, I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $$c^4$$ in the denominator. the 8πG term can be obtained from Poisson's equation, but how does c^4 pop up? Most of the books just derive it...
  29. S

    Question about reverse tracing the Einstein field equations

    From what I know, to get the reverse trace form of the Einstein field equations, you must multiply both sides by gab (I didn't have a lot of time to make this thread so I did not spend time finding the Greek letters in the latex). This turns: Rab- \frac{1}{2}gabR= kTab (where k=...
  30. S

    Manipulation within the Einstein Tensor in Einstein field equations

    Hello everybody. I was recently brainstorming ways to make the Einstein field equations a little easier to solve (as opposed to having to write out that monstrosity of equations that I started on some time ago) and I got an interesting idea in my mind. Here, we have the field equations...
  31. Greg Bernhardt

    What are the Einstein field equations

    Definition/Summary The Einstein Field Equations are a set of ten differential equations which express the general theory of relativity mathematically: they relate the geometry (the curvature) of spacetime to the energy/matter content of spacetime. These ten differential equations may be...
  32. J

    Einstein field equations and scale invariance

    Hi, Are Einstein's field equations without the cosmological constant scale invariant? If so does the addition of the cosmological constant break the scale invariance? John
  33. pellman

    8*pi in the Einstein field equations?

    A typical formulation of the Einstein equations is R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu} The \frac{G}{c^4} make the units work out. What about the 8*pi? Why is this necessary?
  34. T

    Derivation of Einstein Field Equations

    I'm reading Spacetime and Geometry: An Introduction to General Relativity by Sean M. Carrol and in the chapter on gravitation, he derives the Einstein Field Equations. Here is the part I don't get. He starts with the equation R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=\kappa T_{\mu\nu} Wher R_{\mu\nu}...
  35. P

    Linearized Einstein Field Equations

    Hi everyone, Say that one can separate the metric of a space time in a background metric and a small perturbation such that g_{\alpha \beta}=g'_{\alpha \beta}+h_{\alpha \beta}, where g'_{\alpha \beta} is the background metric and h_{\alpha \beta} the perturbation. Computing the christoffel...
  36. A

    Quick question of Einstein Field Equations

    I have seen and read a few different versions of the Einstein field equations (EFE). For example; R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = - 8\piGT_{\mu\nu} , R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8 \pi G}{c^4}T_{\mu\nu} , and 8\piT_{\mu\nu} = G_{\mu\nu} So which one is...
  37. G

    Einstein Field Equations

    Hi all, I have been trying to solve the Einstein Field Equations for its (0,0) component. So I have got that (c=1) Einstein Tensor (upper,0,0)=8*pi*G*T(upper,0,0) Now, let's see what T (0,0) really is. It is energy density, right? So According to famous E=mc^2 the energy density is the same...
  38. A

    Einstein Field equations for dummies

    Hi all! When we talk about the Einstein Field equations. What do we mean with "extremal proper time" or "extremal path"? Why "extremal" ? and why "proper" ? and why do we need to introduce the concept of "geodesic" ? Cheers
  39. Alain De Vos

    Einstein field equations

    Can one deduce from the einstein field equations: -Conservation of mass -Conservation of energy -Conservation of mass-energy -Conservation of linear momentum -Conservation of angular momentum -Principle of least action ? And does curvature of space-time has a "potential" on certain...
  40. E

    An Easy Metric for Einstein Field Equations

    So I am an engineering graduate trying to teach myself some general relativity. I have tried to solve the Einstein Field equations for a wormhole metric and some others. After pages and pages of calculating Christoffel Symbols, Riemann Tensors, Ricci Tensors and Scalars, and so on, I end...
  41. E

    Solving Einstein Field Equations for Minkowski Space with CTC

    For Minkowski spacetime, the metric is: ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 I have read there is a solution when the time dimension is "rolled" into a cylinder forming a closed timelike curve. So the BC is t -> [0,T] with t = 0 identical with t = T. The Field Equation is: Rab - 1/2...
  42. Z

    A few questions about the Einstein field equations?

    1) What exactly does the metric tensor expand into? Since it describes general space-time, shouldn't it be more like a vector like R = √(x^2+y^2+z^2) Why even should we use tensors in relativity when we can just stick with vectors? 2) Are the equations all theoretical? Have they been...
  43. J

    Einstein Field Equations?

    Einstein Field Equations? I have not been able to comprehend the Einstein Field Equation, the Stress Energy Tensor, the Ricci Tensor, the Einstein Tensor, and Christoffel Symbols. Though I am reasonably proficient at working with nested loops in programming, and I have a rudimentary knowledge...
  44. P

    Doubts about Einstein Field Equations

    Why should we trust the Reissner–Nordström solution of charged black holes? It relies on coupling between Einstein tensor and EM stress-energy tensor, which has NO experimental support whatsoever. Is there any chance we can test this?
  45. I

    Pi and Einstein field equations

    I understand the difference betw mathematical and physical pi. I also understand that in non-Euclidean space the value of pi would differ depending on a surface's deviation from flatness. But is there a different symbol for physical pi, to distinguish it from mathematical pi? Because I...
  46. P

    Einstein field equations (EFE's)

    Can someone explain the equation G_{\mu\nu}+\Lambda g_ {\mulnu}={8\ pi G\over c^4} T_{\mu\nu}\ by Albert Einstein?
  47. J

    Linearized gravity / Linearized Einstein Field Equations / GEM

    Are the phrases "Linearized gravity", "Linearized Einstein Field Equations", "GEM (gravitoelectrodynamics)", all referring to mathematically equivalent approximations of Einstein's full non-linear field equations? If not, could someone tell me what order (in some rough sense) these would be...
  48. I

    Exploring the Einstein Field Equations with a Simulation

    Hello, As with a lot of people, I have been excited and fascinated by the field equations Einstein described, revealing the curvature of spacetime. I would like to create a computer simulation which simulates the effects of the Einstein Field Equations, in other words, the curvature of...
  49. M

    Einstein Field Equations Locations

    Looking for Einstein Field Equations, in certain places put that they are 10 and in other places put that they are 16. Which is the correct number?
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