Why Stationary Masses Attract: Explaining GR Intuition

[bryanso]

TL;DR Summary

How does General Relativity explain two stationary objects in free space attract?

In most intuitive explanation of GR there is always a moving object... the moving object travels along the geodesic shortest distance therefore its path is bent toward a massive object.

But let's start with two masses that are stationary in free space. I don't understand how curved spacetime will start to make them move towards each other in the first space...

 

[Ibix]

This is only a problem if you are only thinking of spatial curvature. There is also curvature in the time direction - indeed, neglecting the spatial curvature is part of the simplification process that yields Newtonian gravity. And there's no way to be stationary in time. So, if you remember that the geodesics are paths in the timelike direction, and it's curvature along that direction that's important.

 

[PeroK]

Summary: How does General Relativity explain two stationary objects in free space attract?

In most intuitive explanation of GR there is always a moving object... the moving object travels along the geodesic shortest distance therefore its path is bent toward a massive object.

But let's start with two masses that are stationary in free space. I don't understand how curved spacetime will start to make them move towards each other in the first space...

Your question puzzles me. The simple, intuitive idea is that spacetime is "curved" and objects are influenced to move according the shape of spacetime. The analogy extends to an initially stationary object. Why would a stationary object not respond to a "slope"?

Why do you think objects must initially be in relative motion to attact one another?

 

[Orodruin]

Summary: How does General Relativity explain two stationary objects in free space attract?

the moving object travels along the geodesic shortest distance therefore its path is bent toward a massive object.

Actually, it is more accurate to say that timelike geodesics are stationary paths of proper time. In particular, the straight worldlines of inertial observers in SR maximise proper time between events.

Regardless, the relevant issue is spacetime curvature, tot spatial curvature. The latter also depends on how you define ”space”, which is up to an arbitrary foliation (slicing in spacelike hypersurfaces labelled by a time coordinate) of spacetime. Worldlines are never ”stationary” in spacetime, if they were we would just call them ”events”.

 

[Nugatory]

But let's start with two masses that are stationary in free space. I don't understand how curved spacetime will start to make them move towards each other in the first space...

They are stationary in three-dimensional space but not stationary in four-dimensional spacetime; they are both moving forward in time. Their paths through spacetime are initially parallel but because of curvature draw closer and eventually intersect.

 

[bryanso]

Thank you all. This is very clear, even intuitive.

 

[DaveC426913]

What? Just like that?

That's not how this is done.

You've got to argue.

Oi. Brutus. This guy doesn't want to argue...

 

[A.T.]

I don't understand how curved spacetime will start to make them move towards each other in the first space...

 

[bryanso]

> You've got to argue

Good animation. But how come there is no sound! :)

Thanks!

 

[DaveC426913]

Good animation. But how come there is no sound! :)

Because there is no sound in spaAaAaAaAace...