In case some of you do not understand the acronym, it stands for the Einstein Field Equations. I have them all written down in my notebook and I know basically all of the variables (g for gravity, RGμν is the Einstein Tensor and R for the Ricci Tensor, etc.) I just dont know how to use it in an eqaution.
P.S I am kinda tired at the moment from other things in life which is why I posted in instead of a physics channel.
In: Physics
I don’t know what you’re asking. The Einstein field equations *are* equations (or an equation, depending on your viewpoint). The clue’s in the name. You don’t use the Einstein field equations to solve equations. You solve the Einstein field equations.
Anyway, the question isn’t really ELI5, and neither is the answer. Here’s an exercise for you that might look insultingly trivial: write out the field equations for Minkowski spacetime. So write your metric, work out your Christoffel symbols (which is also insultingly trivial), and from there write out your Einstein tensor. That gives you the symmetries you need in your stress-energy tensor. This should give you a rough idea for the steps you need to set up useful equations.
Then move one step up in complexity: use the metric that gives the line element ds^2=-dt^2 + a(t)^2 d_{ij}dx^idx^j. That’s the Robertson-Walker metric and unlike the Minkowski case, it isn’t insultingly trivial. Your Christoffel symbols exist, and your Einstein tensor is actually interesting. Its symmetries again tell you what you need in your stress-energy tensor, including that now they have to be time-dependent.
What you should end up with are two independent differential equations. (These are known in the jargon as the Friedman equation and depending how you’ve phrased it, a version of the Raychaudhuri equation.) These are redundant with matter continuity (the covariant divergence of the stress-energy tensor).
It’s often more straightforward to use matter continuity in place of an evolution equation given it’s a first-order differential equation. So I’d attack the Friedman equation and matter continuity. To make progress you then make some assumptions about the nature of matter — start off by setting the pressure to zero — at which point you can solve the differential equations.
At the end of that, you’ve got the so-called Einstein-de Sitter universe: a Friedman-Lemaitre-Robertson-Walker universe filled with non-relativistic matter.
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