Proper vs. coordinate acceleration

[Nugatory]

Can there actually be multi-valued paths in GR?

Yes... or to be precise, the result of parallel-transporting a vector in curved spacetime depends on the path along which it is transported. This is a general property of curved manifolds, not unique to GR.

Because comparing two vectors requires parallel-transporting one of them to the other, and there is no unique way of doing this, it is only possible to compare velocities at spatially separated points if they are close enough together that we can treat the spacetime in between as flat.

 

[DrGreg]

And when we say “the second derivative of position” we’re really saying “the second derivative of the spatial coordinates”. Phrased that way, it is more clear that coordinate acceleration is something that is created by our choice of coordinates.

Or, to make it even more clear, “the second derivative of the spatial coordinates with respect to the temporal coordinate”.

 

[Nugatory]

You are walking in the same direction (south) but not along the same path unless you went halfway around at the equator. Nice that there are infinite ways to go south at the pole. This is the source of many fun riddles.

@Orodruin’s example has nothing to do with there being an infinite number of ways to go south at the pole. It works just as well if you start in any direction from any point on the surface of the earth. (The only reason for specifying that you start at the pole is that it's a bit easier to visualize because you're following latitude and longitude lines).

 

[vanhees71]

Fine. Jump off a building and don't worry about because your just in free fall along a geodesic which by definition has no proper acceleration. Good luck with that.

Indeed, that's correct. Jump off the building, and you won't feel much as long as you are in free fall. The trouble only comes, when you hit the ground, but that's not a contradiction to the GR picture of free fall since at the moment you hit the ground you are no longer in free fall but subject to other interactions (mostly electromagnetic) ;-)).

To experience the correctness of the GR picture of free fall rather go to the IRS (rumor has it NASA offers the possibility some time in the future if you are willing to pay the price ;-)) than jumping off your window.

 

[bob012345]

@Orodruin’s example has nothing to do with there being an infinite number of ways to go south at the pole. It works just as well if you start in any direction from any point on the surface of the earth. (The only reason for specifying that you start at the pole is that it's a bit easier to visualize because you're following latitude and longitude lines).

Thanks. What you said is true and I understood the pole wasn't necessary. But it does make for better riddles too.

 

[bob012345]

Indeed, that's correct. Jump off the building, and you won't feel much as long as you are in free fall. The trouble only comes, when you hit the ground, but that's not a contradiction to the GR picture of free fall since at the moment you hit the ground you are no longer in free fall but subject to other interactions (mostly electromagnetic) ;-)).

To experience the correctness of the GR picture of free fall rather go to the IRS (rumor has it NASA offers the possibility some time in the future if you are willing to pay the price ;-)) than jumping off your window.

Thanks, I understand the terminology better now thanks to MTW.

 

[A.T.]

Fine. Jump off a building and don't worry about because your just in free fall along a geodesic which by definition has no proper acceleration. Good luck with that.

There are literally pop songs explaining that:

 

[Ibix]

There are literally pop songs explaining that

Unfortunately, only peer-reviewed songs are acceptable sources for PF.