Undergrad Gravity in General Relativity: Explaining Acceleration

[nikkor180]

Greetings: I can understand that an object's trajectory curves about a greater mass (e. g., satellite in Earth's orbit). The spacetime is curved via the great mass and the moving object simply follows the curvature. My problem is this: Why does a stationary object at a short distance from Earth's surface (or any distance) accelerate toward the center. Can we still explain this via general relativity. How can a curved "track" in space compel a static body to acquire velocity. Is it still the curvature of spacetime that compels a static body to accelerate in magnitude?

 

[Bandersnatch]

Have you seen this animation?:

 

[PeterDonis]

Why does a stationary object at a short distance from Earth's surface (or any distance) accelerate toward the center.

I assume you mean an object that is released from rest at some height above the Earth's surface, so it can free fall downward?

The answer is the same as the one you gave for curving trajectories: the moving object is simply following spacetime curvature.

How can a curved "track" in space compel a static body to acquire velocity.

The trajectory isn't curved in space; it's curved in spacetime.

"Acquire velocity" might be misleading you, because "velocity" is frame-dependent. In a frame in which the Earth is at rest, any object passing near the Earth will, in general, "acquire velocity". That's just as true of an object flying by as of an object released from rest; an object flying by does not just change direction, it changes speed.

 

[PeroK]

Greetings: I can understand that an object's trajectory curves about a greater mass (e. g., satellite in Earth's orbit). The spacetime is curved via the great mass and the moving object simply follows the curvature. My problem is this: Why does a stationary object at a short distance from Earth's surface (or any distance) accelerate toward the center. Can we still explain this via general relativity. How can a curved "track" in space compel a static body to acquire velocity. Is it still the curvature of spacetime that compels a static body to accelerate in magnitude?

The idea of curved spacetime does not mean there are specific tracks in space or spacetime. This seems to be a common misconception. There is another thread about it here, for example:

https://www.physicsforums.com/threads/curved-spacetime-gr-vs-Newtonian-physics.959934/#post-6087645

 

[vanhees71]

Well, freely falling test bodies follow very specific "tracks in spacetime" (shortly called world lines), namely the (time-like or lightlike if you consider massless particles to model "light rays" in the sense of ray optics) geodesics of the spacetime determined by the energy-momentum-stress distribution of the "big objects", e.g., the Schwarzschild solution for a spherically symmetric object.

 

[stevendaryl]

To follow up on what's already been said, here's a picture that might help:

In the absence of gravity, on the left, we have the path of a particle that is sitting at rest at a particular point in space, which means that its path through spaceTIME is a straight line (shown in blue) going parallel to the time axis (in some coordinate system). I'm just show two spatial dimensions to make it simpler. In the presence of gravity, the particle's path through spacetime is no longer a straight line when viewed in the x-y-t coordinate system, but is instead a spiral. That spiral represents an orbit in the x-y plane.

To understand how gravity works, you have to consider that it's not just space that is curved,. but spaceTIME.

 

[kent davidge]

I'm just show two spatial dimensions to make it simpler

And because there's no way to show three spatial dimensions plus time in a two dimensional sheet

 

[PeterDonis]

freely falling test bodies follow very specific "tracks in spacetime"

Yes, but which tracks they follow depend on initial conditions. Flying by the Earth at a fairly high speed is a very different initial condition from being released from rest above the Earth's surface. That's why the two tracks are different. But they both are valid tracks of freely falling bodies in the curved spacetime around the Earth.

 

[vanhees71]

Sure, how a specific body moves in a give spacetime depends on the initial conditions, but that's not very surprising. It's true already in Newtonian physics ;-).

 

[FactChecker]

Have you seen this animation?:

This animation is as good as it gets for visualizing the situation. I think there also is some crude logic that helps:
1) Since we know that spacetime is curved near a massive object, a straight path in it must contain some motion. So a released initially-stationary object must accelerate in some direction.
2) Since the distortion curvature is directly toward the center of the massive object, the acceleration of a released object will either be directly toward or directly away from the massive object.
3) If the acceleration was away from the massive body, there would only be "antigravity" and there would be no massive bodies. (Sorry, I can't think at the moment of any crude mathematical reason for it to be toward rather than away and I can only resort to the non-mathematical evidence.)