Karl Schwarzschild

Schwarzschild's solutions to the Einstein field equations are fundamental to the study of gravitation, as fundamental as Coulomb's law is for electricity. In addition, his research interests were extremely broad, including work in celestial mechanics, observational stellar photometry, quantum mechanics, instrumental astronomy, stellar structure, radiative transfer, stellar statistics, Halley's Comet, and spectroscopy.

Some of his particular achievements include measurements of variable stars, using photography, and the improvement of optical systems, through the perturbative investigation of geometrical aberrations.

Physics of photography

While at Vienna in 1897, Schwarzschild developed a formula, now known as the Schwarzschild law, to calculate the optical density of photographic material. It involved an exponent now known as the Schwarzschild exponent, which is the

p

{\displaystyle p}

in the formula:

i

=

f

(

I

t

p

)

{\displaystyle i=f(I\,t^{p})}

(where

i

{\displaystyle i}

is optical density of exposed photographic emulsion, a function of

I

{\displaystyle I}

, the intensity of the source being observed, and

t

{\displaystyle t}

, the exposure time, with

p

{\displaystyle p}

a constant). This formula was important for enabling more accurate photographic measurements of the intensities of faint astronomical sources.

Electrodynamics

According to Wolfgang Pauli, Schwarzschild is the first to introduce the correct Lagrangian formalism of the electromagnetic field as

S

=

1

2

∫

(

H

2

−

E

2

)

d

V

+

∫

ρ

(

ϕ

−

A

⋅

u

)

d

V

{\displaystyle S={\frac {1}{2}}\int (H^{2}-E^{2})\mathrm {d} V+\int \rho (\phi -\mathbf {A} \cdot \mathbf {u} )\mathrm {d} V}

where

E

,

H

{\displaystyle \mathbf {E},\mathbf {H} }

are the electric and applied magnetic fields,

A

{\displaystyle \mathbf {A} }

is the vector potential and

ϕ

{\displaystyle \phi }

is the electric potential.

He also introduced a field free variational formulation of electrodynamics (also known as "action at distance" or "direct interparticle action") based only on the world line of particles as

S

=

∑

i

m

i

∫

C

i

d

s

i

+

1

2

∑

i

,

j

∬

C

i

,

C

j

q

i

q

j

δ

(

‖

P

i

P

j

‖

)

d

s

i

d

s

j

{\displaystyle S=\sum _{i}m_{i}\int _{C_{i}}\mathrm {d} s_{i}+{\frac {1}{2}}\sum _{i,j}\iint _{C_{i},C_{j}}q_{i}q_{j}\delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)\mathrm {d} \mathbf {s} _{i}\mathrm {d} \mathbf {s} _{j}}

where

C

α

{\displaystyle C_{\alpha }}

are the world lines of the particle,

d

s

α

{\displaystyle d\mathbf {s} _{\alpha }}

the (vectorial) arc element along the world line. Two points on two world lines contribute to the Lagrangian (are coupled) only if they are a zero Minkowskian distance (connected by a light ray), hence the term

δ

(

‖

P

i

P

j

‖

)

{\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)}

. The idea was further developed by Hugo Tetrode

and Adriaan Fokker in the 1920s and John Archibald Wheeler and Richard Feynman in the 1940s and constitutes an alternative but equivalent formulation of electrodynamics.

Thermal radiation

In 1906, Schwarzschild developed the concept of radiative equilibrium between convection inside the Sun and thermal radiation at the surface. He developed an equation for radiative transfer and proved that, in accordance to data, the Sun’s photosphere is in radiative equilibrium.

Relativity

Einstein himself was pleasantly surprised to learn that the field equations admitted exact solutions, because of their prima facie complexity, and because he himself had produced only an approximate solution. Einstein's approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury. There, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant "polar-like" coordinate system and was able to produce an exact solution which he first set down in a letter to Einstein of 22 December 1915, written while he was serving in the war stationed on the Russian front. He concluded the letter by writing: "As you see, the war is kindly disposed toward me, allowing me, despite fierce gunfire at a decidedly terrestrial distance, to take this walk into this your land of ideas." In 1916, Einstein wrote to Schwarzschild on this result:

I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation.

Schwarzschild's second paper, which gives what is now known as the "Inner Schwarzschild solution" (in German: "innere Schwarzschild-Lösung"), is valid within a sphere of homogeneous and isotropic distributed molecules within a shell of radius r=R. It is applicable to solids; incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas.

Schwarzschild's first (spherically symmetric) solution does not contain a coordinate singularity on a surface that is now named after him. In his coordinates, this singularity lies on the sphere of points at a particular radius, called the Schwarzschild radius:

R

s

=

2

G

M

c

2

{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}

where G is the gravitational constant, M is the mass of the central body, and c is the speed of light in vacuum. In cases where the radius of the central body is less than the Schwarzschild radius,

R

s

{\displaystyle R_{s}}

represents the radius within which all massive bodies, and even photons, must inevitably fall into the central body (ignoring quantum tunnelling effects near the boundary). When the mass density of this central body exceeds a particular limit, it triggers a gravitational collapse which, if it occurs with spherical symmetry, produces what is known as a Schwarzschild black hole. This occurs, for example, when the mass of a neutron star exceeds the Tolman–Oppenheimer–Volkoff limit (about three solar masses).

The modern interpretation of the Schwarzschild solution as describing a black hole developed only gradually after Schwarzschild's death. In his original 1916 paper, Schwarzschild used a radial coordinate that differs from the standard Schwarzschild radial coordinate used in modern presentations, and he did not identify the surface now associated with the Schwarzschild radius as an event horizon. The apparent singularity at this radius subsequently became the subject of considerable debate among relativists, including Albert Einstein and David Hilbert. It was only later, through developments in the understanding of coordinate singularities and the extension of the Schwarzschild spacetime, that the surface was recognised as a horizon rather than a physical singularity. This historical development helped establish the Schwarzschild solution as the simplest exact model of a black hole in general relativity.