Table of Contents

## How are Schwarzschild geodesics used in the real world?

Schwarzschild geodesics pertain only to the motion of particles of infinitesimal mass , i.e., particles that do not themselves contribute to the gravitational field. However, they are highly accurate provided that is many-fold smaller than the central mass , e.g., for planets orbiting their sun.

## How is radial free fall related to Schwarzschild equation?

We notice that the two expressions are almost equivalent, except for the presence of the extra term in (1/h3) in the context of the Schwarzschild equation. Radial free fall implies the object is moving ‘straight down’, ie Φ is constant, therefore dΦ/dλ = 0 and so h = r2(dΦ/dλ) = 0

## How did the Schwarzschild metric get its name?

{ extstyle m_ {2}} . This is important in predicting the motion of binary stars in general relativity. The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about a month after the publication of Einstein’s theory of general relativity.

## When does the ratio of Schwarzschild and Newtonian become large?

The ratio only becomes large close to ultra-dense objects such as neutron stars (where the ratio is roughly 50%) and black holes . Comparison between the orbit of a test particle in Newtonian (left) and Schwarzschild (right) spacetime; note the apsidal precession on the right.

## Is the Schwarzschild solution a spherical solution?

According to Birkhoﬀ’s theorem the Schwarzschild solution is also a unique spherical symmetry solution of the vacuum Einstein equation. By spherical symmetry we mean that there is a set of three Killingvectorswith following commutation relations, % V(1),V(2) & = V(3)(5.3) % V(2),V(3) & = V(1)(5.4) % V(3),V(1) & = V(2).

## Is the Schwarzschild solution written as the gravitational constant?

The Schwarzschild solution can be written as is the gravitational constant. The classical Newtonian theory of gravity is recovered in the limit as the ratio goes to zero. In that limit, the metric returns to that defined by special relativity. In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius