This document contains comments about the article Two-body problem in general relativity /A> in Wikipedia
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• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.

### 1.4 Einstein's theory of general relativity

Einstein used a more general geometry, pseudo-Riemannian geometry, to allow for the curvature of space and time that was necessary for the reconciliation; after eight years of work (1907–1915), he succeeded in discovering the precise way in which space-time should be curved in order to reproduce the physical laws observed in Nature, particularly gravitation.
The last part of this sentence should be: he succeeded in discovering the precise way i.e laws, etc, in order to reproduce the same physical behaviour as observed in Nature. The question is exactly which physical processes did he study.

### 2. General relativity, special relativity and geometry

In his special theory of relativity, Albert Einstein showed that the distance ds between two spatial points is not constant, but depends on the motion of the observer.
Exactly which experiment did he describe. The experiment as such has nothing to do with SR. The explanation involving SR is what counts.
However, there is a measure of separation between two points in space-time — called "proper time" and denoted with the symbol dt — that is invariant; in other words, it doesn't depend on the motion of the observer.
If you start from one frame, and you use a grid, than the distance between two points (0,0,0) and (1,1,1) is idependent of t and the speed of an observer
This formula is the natural extension of the Pythagorean theorem and similarly holds only when there is no curvature in space-time. In general relativity, however, space and time may have curvature, so this distance formula must be modified to a more general form
c^2*dT^2 = gµv * dx^µ * dx^v

### 4.1 Post-Newtonian expansion

In the Schwarzschild solution, it is assumed that the larger mass M is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass m follows a geodesic path through that fixed space-time.
As such this is a very simple situation.
The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the post-Newtonian approximation or numerical approximations.
This is the case for all more complex situations.

### 4.2 Modern computational approaches

If there is no incoming gravitational radiation, according to general relativity, two bodies orbiting one another will emit gravitational radiation, causing the orbits to gradually lose energy.
The consequence is that the two masses should spiral together and merge.
The problem is how is this energy loss observed i.e. measured.

### Reflection 1 - General

The two body problem, which involves two masses, is a too simple situation to compare the behaviour of masses using either Newton's Law or the Einstein field equations (i.e. GR) based on actual observations.

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Created: 31 July 2021

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