4
$\begingroup$

I'm a math undergraduate student with some interest in mathematical physics with basic knowledge of partial differential equation.

When I was reading a wikipedia article about einstein field equation,it said

when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.

my question is if einstein equation is a partial differential equation, why can't you solve it normally,why do you need tensor analysis/riemannian geometry for, and can any partial differential equation be written using the languange of tensor, differential geometry,etc?

I apologize for my minimal understanding of this subject, but I haven't learn any tensor calculus yet

$\endgroup$
1
  • $\begingroup$ You can. But if you solve it, you only get a local solution. $\endgroup$ – Si Kucing Aug 15 '19 at 11:36
3
$\begingroup$

Ten is simply the number of distinct components of a second order symmetric tensor in a space of dimension four. Writing these equations in tensor form enables us to write EFE as a single equation instead of ten, just as $\vec F=m\ddot{\vec x}$ is a single vector equation written in place of three scalar equations.

$\endgroup$
6
  • 1
    $\begingroup$ This does not explain why it is formulated in curved space. $\endgroup$ – Si Kucing Aug 15 '19 at 12:42
  • $\begingroup$ I still don't understand any tensor calculus, but can you treat einstein field equation like the usual pde, and can any pde be written and solved in tensor form, if so what is the difference if you use usual method to solve pde? thanks $\endgroup$ – physics noob Aug 15 '19 at 12:43
  • $\begingroup$ I'm afraid I don't understand what you want to know: where did you read of a tensor method to SOLVE pde's? Writing a system of pde's in more compact form can help to find some shortcuts or to give ideas about the solutions, but if you are to solve them (numerically, for instance), then the methods are the usual ones. $\endgroup$ – Intelligenti pauca Aug 15 '19 at 12:58
  • $\begingroup$ Instead, tensor analysis is crucial to understand how EFE were derived (i.e. to understand the physics behind those equations), but has little to do with solving them. $\endgroup$ – Intelligenti pauca Aug 15 '19 at 13:00
  • 1
    $\begingroup$ That's because, in Einstein's view, a free-falling body (e.g. a planet) moves along a geodesic line in space-time. But a space-time with geodesic which are not straight must be curved: Einstein related this curvature to the density of energy-momentum and to do that all the machinery of curved manifolds is needed. $\endgroup$ – Intelligenti pauca Aug 15 '19 at 14:18
1
$\begingroup$

Since you know Einstein field equations, then you must have heard of metric tensor and curvature tensor and their relationship to the field equations.

In local coordinates (normal coordinates) we can make the first derivative of the metric tensor to vanish at a given point but not the second derivative simultaneously. But curvature tensor is a function of both first and second derivatives of the metric tensor, so local coordinates doesnt always help in finding global solutions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.