Archived Pages from 20th Century!!

K. Schwarzschild (1873-1916) |

Just months after Einstein published his work on his Theory of Gravitation,
Karl Schwarzschild (1916) found one solution to Einstein's equations: the
curvature due to a massive nonrotating spherical object. That is, using
Einstein's equation, Schwarzschild had determined how spacetime is curved
due to the presence of a nonrotating spherical mass. In practical terms,
**the Schwarzschild spacetime describes the gravitational field of the
Sun,** or of the Earth. (The Sun and the Earth do rotate, but this rotation
is negligible in these cases.)

This spacetime was studied carefully, and it led to a few **physical
predictions.**

Firstly, it did as good a job as Newton's Theory of Gravity in explaining the motion of the planets around the sun.

Second, it accounted for a tiny effect concerning the path of the planets ("The Anomalous Advance of the Perihelion") that Newton's Theory was unable to completely account for. The orbit of Mercury was studied, and the prediction was confirmed.

Thirdly, it predicted a value for a tiny effect concerning the path of light-rays ("The Bending of Starlight") that Newton's Theory was unable to completely account for. Light from a star passing near the sun was studied. The Einstein Theory corrected predicted the deflection of starlight. (For practical purposes, one could only make the observation during a solar eclipse since sunlight was much brighter than the starlight to be studied.)

**Einstein's Theory (with Schwarzschild's Spacetime) was successful.**

The Schwarzschild Black Hole

Astrophysicists studying stars had determined that the end of a star's
life occurs when the star has exhausted all of its nuclear fuel. In its
death, the star could collapse to form a *black hole* if it were massive
enough.

The spacetime of a **black hole** is **curved** in such a way
as ** to cause the future light cones to tip inward.** At a specific
distance from the black hole, the light cones are so tipped-over that the
"outgoing edge" of each light cone is vertical in the diagram
below. These "edges" form a surface (drawn as a cylinder in the
diagram). This surface (called the

from Penrose, (Scientific American)

The **event horizon** is a boundary that divides this spacetime into
an "inside" and an "outside". Once inside, particles
and light-rays can never escape outside. In fact, since all of the light
cones point to the **singularity** (a really bad place), their worldlines
will end. *(Realize that the light cones restrict the fates of worldlines
that encounter them.)*

The following diagram shows an **observer's worldline** in the outside
region. This observer is periodically sending out light-pulses. The light-pulses
could be detected by, say, another observer who ventures into the black
hole.

from Geroch, (General Relativity from A to B)

The following diagram shows a **foolish observer's worldline** **in
the outside region,** venturing into the black hole. This observer is
periodically sending out light-pulses. However, notice that the closer
our foolish observer gets, the longer it takes for his pulses to reach
an outside observer. Before he reaches the event horizon, the observer
can still return to the outer regions of the outside region... but the
longer he waits, the longer it will take him to return.

**Just after the foolish observer crosses the event horizon** (at
event u), his light-pulses never reach an outside observer. And since his
light-pulses can't reach the outside, no particle (for example, his spaceship)
can reach the outside.

Now, **once inside,** the light cones now direct him to the singularity.
His life will soon be over: his worldline will end.

from Geroch, (General Relativity from A to B)