Schwarzschild geodesics

In general relativity, Schwarzschild geodesics describe how small objects move in the gravitational field of a large, fixed mass. These paths show how things like planets and light move when they are near a big object such as a star or a black hole.

Schwarzschild geodesics helped prove that Einstein’s theory of general relativity was correct. For example, they explain why planets in our Solar System move slightly differently than we would expect, and they show how light bends when it passes close to a massive object.

Schwarzschild geodesics are used when the object moving is much smaller than the mass it is orbiting. This is true for planets going around their stars, where the planet’s mass is tiny compared to the star. These geodesics can also help us understand how two objects, like binary stars, move around each other when we treat the total mass as one big object. This makes it easier to predict their paths in space using Einstein’s ideas about gravity.

Historical context

The Schwarzschild metric is named after Karl Schwarzschild, who discovered it in 1915, just a month after Einstein's theory of general relativity was published. It was the first exact solution of Einstein's equations besides simple flat space, described by the flat space solution.

Later, in 1931, Yusuke Hagihara showed that the path of a small object in the Schwarzschild metric could be described using special math called elliptic functions. Then, in 1949, Samuil Kaplan found that there is a smallest safe distance for a circular orbit around a massive object, called the circular orbit.

Schwarzschild metric

See also: Schwarzschild metric and Deriving the Schwarzschild solution

The Schwarzschild metric is a way to describe space around a round, still object, like a star or a planet. It helps scientists learn how gravity works in space, using Einstein's idea of general relativity.

This metric shows how time and space change close to a big object. For normal things, like Earth or the Sun, these changes are tiny. But for very dense objects, like neutron stars or black holes, the changes get much bigger.

Orbits of test particles

The motion of small objects around a big, fixed mass can be studied using the Schwarzschild metric. This metric helps us understand gravity in space-time. Because of symmetry, we can look at orbits in one plane, which makes things easier.

Two important values that stay the same — energy and angular momentum — help us figure out how these objects move. These values let us predict paths like orbits, spirals, or straight lines falling toward the mass. For example, planets follow stable orbits, while other particles might spiral inward or move in circular paths at certain distances from the central mass.

Precession of orbits

The motion of objects around a large mass, like a planet orbiting the Sun, doesn't follow a perfect circle. Instead, the orbit slowly shifts over time. This shift is called precession.

General relativity explains this shift using something called Schwarzschild geodesics. These show how gravity bends the path of objects around a big mass. One famous example is the planet Mercury. Its orbit moves in a way that older theories couldn't fully explain. General relativity matches what we see.

Bending of light by gravity

See also: Gravitational lens and Shapiro delay

When light passes close to a big object, its path bends because of gravity. This effect is called gravitational lensing. It was predicted by Einstein's theory of general relativity. For light that just touches the Sun, this bending is about 1.75 arcseconds—a tiny but measurable change in the light's direction.

The bending occurs because big objects change the space around them, which affects how light moves. This discovery helped prove Einstein's ideas and is still important for studying stars and galaxies today.

Relation to Newtonian physics

The way things move near a big mass can be explained using ideas from regular physics and Einstein's theory of relativity. For example, when planets go around the Sun, the rules are almost the same as what we learn in basic science — objects are pulled toward the center by gravity, but they also spin around, which keeps them in steady paths.

But Einstein’s theory adds something more. It says that orbits aren’t perfect circles or ovals. Instead, they slowly change shape over time. This change is called precession. It happens because relativity includes an extra force that normal physics does not. We have seen this effect in our solar system, and it helps prove Einstein’s ideas are right.

Mathematical derivations of the orbital equation

The study of Schwarzschild geodesics helps us understand how objects move in strong gravity, such as near a black hole. These geodesics are paths that particles follow in space-time.

One important part is the use of Christoffel symbols. These symbols show how space-time curves because of gravity. They come from the Schwarzschild metric, which describes the space-time around a round mass.

To find the exact paths of particles, scientists use the geodesic equation. This equation shows how particles move when only gravity affects them. By solving this, we can predict how objects orbit big bodies like stars or planets. This helps explain things like light bending near the Sun and the orbits of planets.