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# Experimental evidence of general relativity

As we have seen previously, there are some difficulties to define precisely what a measurement means in general relativity, so as much as possible, we'll try to list the assumptions we're making in doing so.

## General relativity in the classical limit

A simple case of general relativity is the classical limit, in which case we consider the following limits :

1. The metric is close to Minkowski space, $g = \eta + h$, $\| h\| \ll 1$, and we'll ignore any terms in $\mathcal{O}(h^2)$
2. The stress-energy tensor is such that $|T_{00}| \gg |T_{0j}|$, $|T_{ij}| \ll 1$ and all velocities involved are such that $v \ll c$

From the linear expansion around Minkowski space, we know that the first condition implies, given the proper gauge, and defining

$$\gamma_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h$$

we get the linearized gravity equation

$$\Box \gamma_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}$$

...

$$\Box \phi = -\frac{16\pi G}{c^4} T_{00}$$

with solution

$$\phi = -\frac{16\pi G}{c^4} T_{00}$$

### Tests of free fall

The case of free fall can be done with slightly different assumptions from the case of the Newtonian limit, although both will lead us to the same equations.

Consider the following assumption :

1. The spacetime metric around Earth can be described by a Schwarzschild metric
2. Any object in free fall we consider can have its trajectory computed in the test particle limit

The (exterior) Schwarzschild metric is, as usual

$$ds^2 = -(1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r}) dr^2 + r^2 (d\theta^2 + \sin(\theta) d\varphi^2)$$

The simplifying assumptions involved here are that the Earth's stress-energy tensor is isotropic around its center and static, as well as having a negligible rotation. We also assume that regions beyond the surface of the Earth can be approximated by a vacuum, and that the influence of far off objects can be neglected.

## The special relativistic limit

Another fairly simple limit is the non-interactive limit, $G \to 0$. Assuming otherwise Minkowski space, this leads to the test of special relativity.

## Gravitational wave detection

Last updated : 2023-05-08 14:42:20