# Is an orbiting object traveling along a geodesic in general relativity?

I am getting a layman's understanding of General Relativity. I understand that gravity is understood to be an objects propensity to travel along a straight line in curved spacetime. Would an orbiting object be considered to be traveling along a geodesic if it has its own velocity and falls into an orbit, or would geodesic motion only apply to something like free-fall? What is the difference between an orbit of an object with mass and the bending of light? Is it only light that travels along the geodesic?

Would an orbiting object be considered to be traveling along a geodesic if it has its own velocity and falls into an orbit, or would geodesic motion only apply to something like free-fall?

Orbit is free-fall, unless you're assuming that there are non-gravitational influences at work (e.g. a rocket engine). Geodesics are the trajectories followed by objects in the absence of non-gravitational influences.

What is the difference between an orbit of an object with mass and the bending of light? Is it only light that travels along the geodesic?

Light follows null geodesics whose tangent vectors have zero norm, while massive objects in free-fall follow timelike geodesics whose tangent vectors have negative norm if your metric signature is $$(-+++)$$ and positive norm if your signature is $$(+---)$$.

• So geodesic motion is any motion where there is no acceleration from the internal reference frame, e.g. an orbit or falling directly towards the earth. Is this correct? Feb 14, 2021 at 6:17
• @sakurashinken Yes. Basically, if you would in classical mechanics consider gravity to be the only force that affects the motion of your object, then in general relativity that corresponds exactly to geodesic motion. Feb 14, 2021 at 9:48

In physics the expression 'free fall' is used in a wider sense than in everyday language.

In everyday language when you say 'free fall' you will generally mean falling straight towards the center of gravity.

In physics conversation a more general meaning is used. When an object has a velocity component perpendicular to the direction of gravity the motion is still put in the general category 'free fall'.

Now take a satellite in orbit. That satellite does accelerate toward the center of gravity, but it has so much velocity component perpendicular to the direction of gravity that the satellite never makes it to the surface. Being in orbit is that you are falling all the time, but you never make it to the surface.

In physics conversation the meaning of 'free fall' extends even to orbiting motion.

The 'free' in the 'free fall' is thought of as referring to the fact that an onboard accelerometer will register zero acceleration. I'm specifically referring to an onboard accelerometer. The accelerometer is of the type that it measures whether some propulsion is pulling G's. A satellite in orbit isn't pulling G's.

The concept of a geodesic does not refer to something that is inherent in the spacetime.

An orbiting satellite is moving along a geodesic. Let this satellite be of a type that it also has some strong onboard propulsion available, so that the satellite can change its own velocity.

After a burst of propulsion the velocity of the satelite has been changed and it is subsequently in an orbit that has a different shape than the previous orbit. Whatever the origin of the velocity: every orbit is a geodesic. The criterium is that the object in motion isn't pulling G's.

For any object the shape of the geodesic that it is moving along follows from both the distribution of gravity and the velocity of the object with respect to that distribution of gravity.

In the presence of gravity the path that light follows is far less curved than the path of slower moving entities, because light moves so much faster. Light moves along the geodesic that corresponds to the geodesic for that velocity of propagation.

The answer to your question is yes. Just think about the ISS, and the astronauts. Why do they feel weightless? Contrary to popular belief, it is not because the only force acting on them is gravity. In reality, there are multiple forces acting on them, but they all cancel out to zero.

There is another very nice example of this with the cannonball.

If the speed is the orbital speed at that altitude, it will go on circling around the Earth along a fixed circular orbit, just like the Moon. (C) for example horizontal speed of at approximately 7,300 m/s for Earth.

https://en.wikipedia.org/wiki/Newton%27s_cannonball

You see some people say that the astronauts on the ISS are floating because they are in a zero gravity environment, where spacetime is flat. In reality, the gravitational force (acceleration) is 90% at the altitude of the ISS compared to here on the surface of Earth.

on the ISS for example the force of gravity is about 88% the value at the Earth's surface

Cause of weightlessness

What happens to the ISS is that there is another pseudo force (centrifugal force, their velocity) pointing in a direction away from Earth, and that cancels out gravity to a net zero. So the astronauts truly feel weightless, because the net force acting on them cancels out to zero. They are in free fall.

The centrifugal force counteracts the earth's gravitational force of attraction.

But are they moving along a geodesic? The answer is a sound yes according to GR.

a freely moving or falling particle always moves along a geodesic.

https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

Now you are asking about light, and I believe you are asking about gravitational lensing, and yes, light moves around the massive object along a geodesic, we just happen to call that a null geodesic.