Difference between General and Special relativity

whereas in general relativity you can have f(x,t) in front of $c^2dt^2, dx^2, dy^2, dz^2$ depending on the gravitational object you're dealing with.

I know that general relativity applies in non-inertial reference frames whereas special relativity does not but what difference does this make? does the math of special relativity work in the general relativity. if not what is the difference?

also does the theory hold in a reference frame undergoing NON-UNIFORM acceleration?

General Relativity works for everything.

Just saying, people tend to use the form $c^2 d\tau^2 = g_{\mu\nu}dx^\mu dx^\nu$, where $g_{\mu\nu}$ denotes the metric describing the space-time, because if you use $c^2 d\tau^2=A dt^2 - B dx^2 - C dy^2 - D dz^2$, you imply the metric tensor is diagonal, which is not really the case for most metric, for instance, Kerr metric.

General Relativity works for everything. If there is no acceleration, the it simplifies to Special Relativity .

The maths of Gen Rel will work in Spec Rel (it'll just simplify to Spec Rel maths), but the opposite is not true...

It works for everything about gravity.

General relativity describes gravity via geometry of space-time.

As a supplementary note, general relativity becomes special relativity at the limit of weak field in which one can do the following approximation:

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu \nu}$ and that $||h_{\mu\nu}||\ll1$

In words:

A metric $g_{\mu\nu}$ describes space-time and the metric in a flat space-time aka Minkowski metric is the metric in special relativity which is



The really awesome feature of this metric is that it is constant, so maths will be really simple, and that's also the reason why the maths in SR is much simpler than in GR.

On the other hand, $h_{\mu \nu}$ is the perturbation term, which is used to alter the original Minkowski metric. $||h_{\mu \nu}||\ll 1$ means the "magnitude" of the metric is really small so that it can still be approximated as a linear field.

Actually, it should be:

"If there is no gravity, then it simplifies to Special Relativity"


Special Relativity can handle pure electromagnetic accelerations perfectly well. It's only when you have gravitational accelerations that the curvature of spacetime needs to be accounted for.

Ah, okay . Thanks.

No problem. The only caveat is that the electromagnetic force is caused by photons and photons have energy and energy curves spacetime. So having a pure electromagnetic force is technically impossible. There will always be (at least) a tiny contribution from gravity.