# What is field equations: Definition and 105 Discussions

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.
A physical field can be thought of as the assignment of a physical quantity at each point of space and time. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change.
The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics before the advent of relativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic. Modern field theories are usually expressed using the mathematics of tensor calculus. A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles.

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1. ### I Numerical Relativity -- Software to solve the Einstein field equations?

Hello, Has anyone here used software to solve the Einstein field equations? If so, what software have you used, or what software do you recommend? Is it possible to use something like MATLAB to play around with the field equations?
2. ### A Extra (boundary?) term in Brans Dicke field equations

Here is the action: ##S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} (R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b} + 16\pi L_m)## the ordinary matter is included via ##L_m##. Zeroing the variation ##\delta/\delta g^{\mu \nu}## in the usual way gives ##\frac{\delta}{\delta g^{\mu \nu}}[R\phi...
3. ### A Derivation of the Maxwell and Proca Field Equations

I am back to my writing desk and I was looking up different and (hopefully) relatively "basic" derivations of the field equations. I found a nice little derivation of the Proca (and Maxwell) equations by Gersten (1998, PRL, 12, 291-298 is the reference, but I got it from the web so what I have...
4. ### A The field equations of elasticity

First, my ignorance... I know there are classes of equations: Laplace, Poisson, Wave, Diffusion, etc. (I suppose Laplace is a subset of Poisson, but that is not the issue). Into what category of mathematical equations would you place the field equations of elasticity (stress/strain/displacement)?
5. ### I Solving the EM field equations to produce the desired vector field

So, we have A, the magnetic vector potential, and its divergence is the Lorenz gauge condition. I want to solve for the two vector fields of F and G, and I'm wondering how I should begin##\nabla \cdot \mathbf{F}=-\nabla \cdot\frac{\partial}{\partial t}\mathbf{A} =-\frac{\partial}{\partial...
6. ### 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...
7. ### 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...

9. ### I Einstein Field Eqns: East/West Coast Metrics

My questions is: Depending on which metric you choose "east coast" or "west coast", do you have to also mind the sign on the cosmological constant in the Einstein field equations? R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \pm \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} For example, if you...
10. ### I Exploring Physical Connections in Einstein's Field Equations

While viewing a recent lecture on Einstein’s Field Equations, the presenter made the association between the various quantities like mass, energy, momentum and pressure directly to the four dimensions of spacetime. Depending on which derivation of the FE he was explaining, he would make an...
11. ### 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?
12. ### 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...
13. ### Insights Solving Einstein's Field Equations in Maxima - Comments

Greg Bernhardt submitted a new PF Insights post Solving Einstein's Field Equations in Maxima Continue reading the Original PF Insights Post.
14. ### A Solving Schwarzschild Field Equations in this Form

Applying Cartan's first and second structural equations to the vielbein forms \begin{align} e^t = A(r) dt , \ \ \ \ \ e^r = B(r) dr , \ \ \ \ \ e^{\theta} = C(r) d \theta , \ \ \ \ \ e^{\phi} = C(r) \sin \theta d \phi , \end{align} taken from the metric \begin{align} ds^2 = A^2(r) dt^2 - B^2(r)...
15. ### 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?
16. ### B Exploring Einstein's Field Equations: Space Density

Hi, I'm trying to understand Einstein's field equations conceptually, does it describe space density in a region of space by any chance? Like there is more space in this region compared to this other region. Thanks.
17. ### 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)...
18. ### A Multipole expansion of linearized field equations

I read Chris Hirata's paper on gravitational waves (http://www.tapir.caltech.edu/~chirata/ph236/lec10.pdf) where he performs a multipole expansion of the gravitational source. I got most of it, apart from the part where he expands the inverse distance function into a series : More specifically...
19. ### 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...
20. ### B Solving Field Equations & Schwarszchild Metric

I have read that Albert Einstein was quite (pleasantly) surprised to read Schwarzschild's solution to his field equation because he did not think that any complete analytic solution existed. However, of all the possible scenarios to consider, a point mass in a spherically symmetric field (ie, a...
21. ### 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?
22. ### I Should Einstein's Field Equations be modified for cosmology?

Let us consider the FRW metric for flat space expressed in terms of conformal time ##\eta## and cartesian spatial co-ordinates ##x,y,z##: $$ds^2=a^2(\eta)\{d\eta^2-dx^2-dy^2-dz^2\}.$$ As in the standard FRW co-ordinate system one can see that if two observers are separated by a constant...
23. ### I Solving the Gravitational Field Equations

I have read that: In 1915 Einstein presents to the Prussian Academy of Sciences the General Theory of Relativity; it includes a set of Gravitational Field Equations; at this time he does not present any solution to the equations. In 1917 he considers a greatly simplified case; presents a...
24. ### 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.
25. ### I Field equations fully written out

Hi, Does anybody know a link where the Einstein field equations are fully written out, i.e. in terms of only the coefficients of the metric tensor and derivatives on the left side? I'm just curious how huge this must be.
26. ### Obtaining field equations from an action

Homework Statement Provided an action: S[A_\nu] = \int\left(\frac{1}{4}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta A_\mu J^\mu\right)\sqrt{-\eta}~d^4x How would one go about finding the field...
27. ### I Dimension Check of Einstein's Field Equations

This is my first post in Physics Forums. I am trying to a dimensional check of Einstein's field equations. Unfortunately, most books consider c = 1 or sometimes even G = c = 1, when presenting the field equations. This makes it very difficult to do a dimensional check. In spite of this, I...
28. ### 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...
29. ### Einstein-Cartan Field Equations Derivation

I'm looking for a derivation of the Einstein-Cartan Equations from varying the Einstein-Hilbert Action. I've looked around the internet but can't seem to find one, so if anyone knows of a pdf or book reference that goes through the derivation, that would help a lot. The wikipedia page for...
30. ### 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...
31. ### 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?
32. ### Field equations Einstein-Gauss-Bonnet action

Hello everyone! I have a "great" problem with EGB action. First of all, I'm used to work with potential and scalar field, but now I have the following action ##S=\int\sqrt{-g}\left(2\beta +R+\alpha GB\right)d^6 x## where GB is the six-dimensional Gauss-Bonnet term, R is the scalar curvature...
33. ### Deriving Field Eqns from Gauss-Bonnet Lagrangian

How to derive the field equations from a Gauss-Bonnet Lagrangian?
34. ### Einsteins field equations us what type of index notation?

I know that the metric tensor itself utilizes Einstein summation notation but the field equations have a tensor form so the μ and ν symbols represent tensor information. I'm trying to wrap my head around how Einstein used summation notation to simplify the above field equations but it seems...
35. ### 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
36. ### 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...
37. ### The Field Equations of Newton: Understanding the Basics

Am I right when I say during Newton's time there was no idea of fields? Now I have been looking for books and courses which are meant for amateurs. So I came across this video of one of my favourite professors Prof Leonard Susskind...
38. ### Understanding Einstein's field equations

Hi, I'm interested in trying to understand Einstein's field equations, I'm a physics student due to start an astrophysics course next year. I was just wondering if someone could give me some advice where to start?
39. ### 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=...
40. ### Possibly solved the metric without field equations

Error below :grumpy: For a couple of years now, I have been attempting to solve for the values in GR of the time dilation z and the radial and tangent length contractions, L and L_t respectively, which form the metric c^2 dτ^2 = c^2 z^2 dt^2 - dr^2 / L^2 - d_θ^2 r^2 / L_t^2 (along a plane)...
41. ### 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...
42. ### 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...
43. ### A few questions about Einstein's field equations

1. What exactly is the cosmological constant, what is its value (or how do you derive it if it is something that must be derived for various situations) and how do I know when to use it and when not to use it in the Einstein field equations? I ask this last question because sometimes I see...
44. ### Are Solutions to GR Field Equations Chaotic?

The field equations of general relativity are non-linear. There are exact analytic solutions to the equations for special symmetrical cases, e.g. Schwarzschild's solution for a black hole. But in general, wouldn't there be other chaotic solutions as well? A chaotic system is analytically...
45. ### Einstein's Basis for Equivalence in his Field Equations

The following is a question regarding the derivation of Einstein's field equations. Background In deriving his equations, it is my understanding that Einstein equated the Einstein Tensor Gμv and the Cosmological Constant*Metric Tensor with the Stress Energy Momentum Tensor Tμv term simply...
46. ### Electromagnetic field equations of motion

1. I'm not quite sure how the laplacian acts on this integral 2. \frac{\delta S}{\delta A_{\mu}}=\int\frac{\delta}{\delta A_{\mu}}(\frac{1}{4}F_{\rho\sigma}\frac{\triangle}{M^{2}}F^{\rho\sigma}) 3. I know I have to split the integral into three integrals for x y and z, but I'm not sure if a) I...
47. ### 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
48. ### 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?
49. ### Electromagnetic Field Lagrangian - Field Equations

I was working on an exercise in Ohanian's book. [Appendix A3, page 484, Exercise 5] I guess he means charge conservation, but wrote ##j^\nu = 0##. The Lagrangian was given by ##\mathcal{L}_{em} = -\frac{1}{16\pi} \left( A_{\mu ,\nu} - A_{\nu ,\mu} \right) \left( A^{\mu ,\nu} - A^{\nu...
50. ### Operator equations vs. field equations

If my understanding is correct, the equations of QFT (Dirac, Klein-Gordon) govern the behavior of operator fields (assigning operator to each point in space). Does it mean there are no equations governing the behavior of fields (assigning a number / vector/ spinor to each point in space)? Is QFT...