I am only about half-way of studying SR and GR and I am not yet familiar with a formula to calculate the curvature of spacetime when energy is present. To be more specific, I want to calculate curvature when electrical energy is present.

Also, I would like to know how energy influences curvature of an object of certain mass charged with electrical energy (e.g. an iron object).

  • 2
    $\begingroup$ You need to work out the energy momentum tensor $T^{\mu\nu}$ that goes on the RHS of Einstein's eq. $\endgroup$
    – Virgo
    May 28 '15 at 9:37
  • $\begingroup$ @physicsphile Alright. Once I have done that, how do I use it in a formula calculating curvature? $\endgroup$
    – John K
    May 28 '15 at 9:44
  • $\begingroup$ as noted above, you include it on the rhs in the einstein equation, then think of all the physical constraints that could reduce the number of coordinate variables and give you the greatest chance of solving the lhs, i.e the curvature tensor. Look at the assumptions made to simplify the standard schwartzchild solution, except you will have a set of inhomogeneous P.D.Es, best of luck.... $\endgroup$
    – user81619
    May 28 '15 at 22:53

As said in the comments, you need to use Einstein's equations (no cosmological constant for simplicity):

$$R_{\mu\nu} - \frac12 R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

Your energy goes into the energy-momentum tensor $T_{\mu\nu}$; in particular, there is a formula which you can use to find the energy-momentum tensor of an electromagnetic field. The left hand side contains $R_{\mu\nu}$ and $R$, which are very complicated functions of the metric tensor $g_{\mu\nu}$ and its derivatives. Since all the tensors here are symmetric, this is a system of 10 coupled nonlinear partial differential equations which you can solve in principle to find the metric tensor.

In practice, almost no one does that. If you think that the curvature will be small then you can get an approximate version of the equation that is linear and relatively straightforward to solve. If you can't do that then you will either need some symmetry to simplify the metric tensor (such as spherical symmetry for the Schwarzschild solution), or solve the equations numerically, which isn't easy either. Wikipedia has some information on the subject.


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