/sci/ - Science & Math

[Anonymous No.9306916]

Can someone explain me how to "use" Einstein's gravity equations to make simple predictions or to get basic values?
I am guessing that one needs to work with the field equations.
For example, how can you get the acceleration experimented by a 1 kg object at rest on the surface of the earth?
Let's assume the Earth radius at that point is of 6370 kilometres. And Earth's mass to be 5.97E+24 kilograms.
When working with Newtonian gravity we could simply use the equation everyone knows:
[math]F=G{\frac {m_{1}m_{2}}{r^{2}}}[/math]

Note that this is no homework. I am simply want to know more about it. And I think that looking at how to solve a problem is a good approach to learning.

[Anonymous No.9306923]

>>9306916
I simply*

[Anonymous No.9306968]

To show you where things come togeter...

If you know electrostatics or the use of potentials f for Newtonian gravity, then you know that 1/r^2 (propto Df) for a vaccum comes about through the Possion equations DDf(r)=q(r), which spefically reads
(1/r^2) D (r^2 Df) = 0

In relativity, f would be part of the the time-component of the metric g and the Einstein tensor G is essentially the Laplace operator DD applied to the metric. Except there are 9 more independent components, so things don't get simpler.

There is a Wikipedia page with exact solutions, but it's mostly not possible.
You also have the derivation here.

https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

But essentially, it's just a differential equation and you got to do a lot of modeling work to get "what you want" out of it. Gödel trolled Einstein quite a bit when he explicitly worked on finding a solution where time goes in circles and such. Not all that pops out of it is relevant physics. There was a phase in the 50's when GR became super popular again, and that's they era where le Hawkings comes from too. But it doesn't matter too much anyway. It was coocked up in the 20 and one two decades later they were already desperately trying to find a quantum version.

[Anonymous No.9306991]

here that meme solution I talked about

https://en.wikipedia.org/wiki/G%C3%B6del_metric

Here's 85 of the more classical elternatives to general relativity, i.e. those after Einstein but before stuff like loop quantum gravity

https://en.wikipedia.org/wiki/Alternatives_to_general_relativity

Einstien tensor

https://en.wikipedia.org/wiki/Einstein_tensor

This table helps too

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

[Anonymous No.9307181]

Differential equations with tensors.
Not familiar with relativity but in a simple case most components should be zero so you get a system of differential equations and you have your answer.

[Anonymous No.9307208]

>>9306916
The problem is that each side depends on each other. So the only way to get solutions is to guess properties and the form of the solution, then see if it matches any observed phenomena. The Schwarzschild metric is one of the simplest. The way that solution is obtained is by assuming many things are zero and that the solution should have symmetries like being spherically symmetric, from these the form of the solution can be obtained.

[Anonymous No.9307636]

>>9306968
>>9306991
>>9307181
>>9307208
Thanks.
Reading all of this; it feels like it is very difficult to do this.
What about computational numerical solutions for n-body problems?
With newtonian gravity, on the most direct approach, one just uses an integrator (euler, verlet, Runge-Kutta...), and "that's all".

Is it really impossible to do this relativistically?

[Anonymous No.9307688]

>>9306916
Einstein's equations are really for when you want to study how the mass-energy-metric interplay determines the geometry of the space time and the evolution of big masses therein.

the problem of a falling body in a central gravitational field is typically done by keeping the metric fixed, as a background (such metric, by the way, is calculated using Einstein's equations, of course, which, for a central gravitational force/mass configuration, have the Schwarzschild metric as solution).

So, fix the metric. The motion of a falling body, not subject to other forces, is described by a geodesic with respect to that metric. The mass of the body plays no role, the only thing that matters is that the space-time worldline of the falling body is a geodesic. So that is how you compute its motion.

[Anonymous No.9308525]

bump