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ⓘ Schwarzschild metric. The Schwarzschild metric was calculated by Karl Schwarzschild as a solution to Einsteins field equations in 1916. Also known as Schwarzsch ..




Schwarzschild metric
                                     

ⓘ Schwarzschild metric

The Schwarzschild metric was calculated by Karl Schwarzschild as a solution to Einsteins field equations in 1916. Also known as Schwarzschild solution, it is an equation from general relativity in the field of astrophysics. A metric refers to an equation which describes spacetime; in particular, a Schwarzschild metric describes the gravitational field around a Schwarzschild black hole - a non-rotating, spherical black hole with no magnetic field, and where the cosmological constant is zero.

It is essentially an equation that describes how a particle moves through the space near a black hole.

d s 2 = − c 2 1 − 2 G M r c 2 d t 2 + 1 − 2 G M r c 2 d r 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d ϕ 2 {\displaystyle ds^{2}=-c^{2}1-{\frac {2GM}{rc^{2}}}dt^{2}+{\frac {1}{1-{\frac {2GM}{rc^{2}}}}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin ^{2}\thetad\phi^{2}}

                                     

1. Derivation

Although a more complicated way of calculating the Schwarzschild metric can be found using Christoffel Symbols, it can also be derived using the equations for escape velocity v e {\displaystyle v_{e}}, time dilation dt, length contraction dr:

v e = v = 2 G M r {\displaystyle v_{e}=v={\sqrt {\frac {2GM}{r}}}} 1

v is the velocity of the particle G is the gravitational constant M is the mass of the black hole r is how close the particle is to the heavy object

d t ′ = d t 1 − v 2 c 2 {\displaystyle dt=dt{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}} 2 d r ′ = d r 1 − v 2 c 2 {\displaystyle dr={\frac {dr}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} 3

dt is the particles true change in time dt is the particles change in time dr is the true distance traveled dr is the particles change in distance v is the velocity of the particle c is the speed of light

Note: the true time interval and true distance traveled by the particle are different than the time and distance calculated in classical physics calculations, since it is traveling in such a heavy gravitational field!

Using the equation for flat spacetime in spherical coordinates:

d s 2 = − c 2 d t 2 + d r 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d ϕ 2 {\displaystyle ds^{2}=-c^{2}dt^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin ^{2}\thetad\phi^{2}} 4

ds is the path of the particle

θ {\displaystyle \theta } is the angle d θ {\displaystyle \theta } and d ϕ {\displaystyle \phi } are the change in angles

Inputting the equations for escape velocity, time dilation, and length contraction equations 1, 2, and 3 into the equation for flat spacetime equation 4, to get the Schwarzschild metric:

d s 2 = − c 2 1 − 2 G M r c 2 d t 2 + d r 2 1 − 2 G M r c 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d ϕ 2 {\displaystyle ds^{2}=-c^{2}1-{\frac {2GM}{rc^{2}}}dt^{2}+{\frac {dr^{2}}{1-{\frac {2GM}{rc^{2}}}}}+r^{2}d\theta^{2}+r^{2}\sin ^{2}\thetad\phi^{2}} 5

From this equation we can take out the Schwarzschild radius r s {\displaystyle r_{s}}, the radius of this black hole. Although this is most commonly used to describe a Schwarzschild black hole, the Schwarzschild radius can be calculated for any heavy object.

d s 2 = − c 2 1 − r s r d t 2 + 1 − r s r d r 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d ϕ 2 {\displaystyle ds^{2}=-c^{2}1-{\frac {r_{s}}{r}}dt^{2}+{\frac {1}{1-{\frac {r_{s}}{r}}}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin ^{2}\thetad\phi^{2}} 6

r s {\displaystyle r_{s}} is the set radius limit of the object