If a body is initially stationary, it falls when released. Why does it fall along a geodesic?

  • #1

Summary:

The path taken by a body in free fall is a geodesic. But if the body is initially stationary it falls when released. Why?
We are told gravity is a curvature in spacetime but the force of gravity only seems to apply if the body is moving. What moves it if it starts off stationary and then falls?
 

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  • #2
Ibix
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We are told gravity is a curvature in spacetime but the force of gravity only seems to apply if the body is moving. What moves it if it starts off stationary and then falls?
They are geodesics in spacetime. You can't not be moving forwards in time.

You are probably being misled by pictures showing space around Earth as curved. In such diagrams, all the interesting stuff happens in the time direction - which isn't shown!
 
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  • #3
A.T.
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But if the body is initially stationary it falls when released. Why?
 
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  • #4
phinds
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Summary:: The path taken by a body in free fall is a geodesic. But if the body is initially stationary it falls when released. Why?
It was only stationary because a force was keeping it from following the geodesic (which is why it had to be released) and when the force is released, it moves along the geodesic.
 
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  • #5
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It was only stationary because a force was keeping it from following the geodesic
In this context, "stationary" means that the object is following a non-geodesic path through spacetime parallel to our own, and therefore is not moving relative to us.
(Phinds knows this of course - this comment is for OP as they read the replies)
 
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  • #6
In this context, "stationary" means that the object is following a non-geodesic path through spacetime parallel to our own, and therefore is not moving relative to us.
(Phinds knows this of course - this comment is for OP as they read the replies)
They are geodesics in spacetime. You can't not be moving forwards in time.

You are probably being misled by pictures showing space around Earth as curved. In such diagrams, all the interesting stuff happens in the time direction - which isn't shown!
I get it. Thanks.
 
  • #7
They are geodesics in spacetime. You can't not be moving forwards in time.

You are probably being misled by pictures showing space around Earth as curved. In such diagrams, all the interesting stuff happens in the time direction - which isn't shown!
Yes I see. Thank you.
 
  • #8
Now that I've thought about your replies it seems to me you haven't really answered my question because although time is a part of spacetime it doesn't curve around a mass like space does, so far as I know. And although the mass moves through time it is presumably not moving along one of the geodesics of space. So what pulls it towards the large mass (earth)? You can't have it both ways: (i) there is no separate space, only spacetime (ii) the geodesic is space curving, not time. Please tell me where this is wrong.
 
  • #9
PeterDonis
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although time is a part of spacetime it doesn't curve around a mass like space does
Yes, it does. The video @A.T. posted is describing curvature of time. More precisely, it's describing effects of spacetime curvature that show up as effects on time, not space.

although the mass moves through time it is presumably not moving along one of the geodesics of space
No, the mass is moving on a geodesic of spacetime. Its path in space is, except in certain special cases like purely radial motion, not a geodesic in space.

You need to stop thinking in terms of separate "time" and "space" and start thinking in terms of spacetime.
 
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  • #10
Ibix
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because although time is a part of spacetime it doesn't curve around a mass like space does
It most certainly does! Newton's theory of gravity is an approximation to general relativity in the case where curvature in the time direction is the only important curvature (wincing slightly as I write that - more precisely ##G^{tt}=0## is the only equation with significant terms). So near Earth, where non-Newtonian behaviour of gravity is incredibly difficult to detect, spatial curvature is more or less irrelevant. Pretty diagrams of such curvature are impossible to draw, however, so all you see outside serious textbooks is the unimportant spatial curvature, seriously exaggerated.

Masses follow geodesics in spacetime. The spatial projection of those paths may or may not be geodesics of the spatial slices (I don't know), but it isn't important for anything.
 
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  • #11
I'll think about it. Thanks for now.
 
  • #12
PeterDonis
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The spatial projection of those paths may or may not be geodesics of the spatial slices (I don't know)
As I said, in all but a few special cases (such as purely radial motion), the spatial projections will not be geodesics of the spatial slices.
 
  • #13
A.T.
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...not moving along one of the geodesics of space...
Correct, the free falling path doesn't have to be a geodesics of space. The spatial path is a projection of the space-time geodesic onto the spatial dimensions.
 
  • #14
Ibix
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As I said, in all but a few special cases (such as purely radial motion), the spatial projections will not be geodesics of the spatial slices.
Sorry - didn't see your post. I suppose it's kind of obvious though. If we're arguing that space is more or less flat near Earth then circular orbits are manifestly not geodesics of "nearly" flat space.
 
  • #15
I am now thinking in terms of spacetime and my comments are now:- Let the smaller masses' projections of spacetime geodesics on to space be called space paths for simplicity in my question and its likewise projection of spacetime geodesics on to time and which I will call time pathways for simplicity in this question, (I) are they the identical? (ii) since the stationary mass I originally proposed is moving in time, as Ibix said, and moving along a projected time geodesic, it is presumably not accelerating in time and so there is no force of gravity acting on it. (iii) since this mass crosses no projected space geodesic it is not accelerating in space so there is no force of gravity acting on it. (iv) why does it fall? (v) where are the fallacies in questions (i) to (iv)? I don't see why I must stop thinking in terms of space and time as separate identities from now on. To do so would prevent me asking more fundamental questions about gravity.
 
  • #16
PeterDonis
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Let the smaller masses' projections of spacetime geodesics on to space
"Space" depends on your choice of coordinates so what you are doing here is taking you away from the actual physics, not towards it. The actual physics is contained in invariants, quantities that are independent of your choice of coordinates.

its likewise projection of spacetime geodesics on to time
There is no such thing; you can't project a curve in spacetime "onto time".

I don't see why I must stop thinking in terms of space and time as separate
Because they're not separate entities in relativity. The entity is spacetime; "space" and "time" depend on your choice of coordinates. If you continue to insist on thinking of them as separate, you will simply not be able to understand the things you are trying to understand.
 
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  • #17
Orodruin
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To do so would prevent me asking more fundamental questions about gravity.
On the contrary. Not to do so would prevent you from understanding anything fundamental avout the description of gravity in terms of general relativity.
 
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  • #18
Ibix
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I don't see why I must stop thinking in terms of space and time as separate identities from now on. To do so would prevent me asking more fundamental questions about gravity.
The fundamental truth is that, at least in relativity, spacetime is one entity, and you can't generally split them up in any meaningful way. There's always some freedom to choose how you define space and time, and if there's freedom for you personally to choose then there can't be any physical significance to the choice. Of course, there are often obvious ways to do the split. And there's always an obvious way for a person to do it for spacetime near themselves - it's just not possible to unambiguously extend that to everything.

General relativity is our most fundamental theory of gravity so far. If you ask questions about gravity, people will answer in terms of spacetime. You may not like those terms, but you will need to accept them to be able to communicate about gravity.

Let the smaller masses' projections of spacetime geodesics on to space be called space paths for simplicity in my question and its likewise projection of spacetime geodesics on to time and which I will call time pathways for simplicity
It makes sense to talk of the projection of geodesics onto space, since space is a 3d slice through spacetime. It's like the shadow of a curved rod on the ground. It doesn't make sense to talk about the projection of the geodesic onto time since that's a 1d slice of spacetime. It's like asking what shape the shadow of the curved rod is on a hair hanging vertically. For a start, you need to specify where the hair is, and the answer is going to be pretty unexciting (it'll look like a bit of hair) once you've done it.
 
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  • #19
Orodruin
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It makes sense to talk of the projection of geodesics onto space, since space is a 3d slice through spacetime.
Technically, this is only true in stationary spacetimes. If your spacetime is not stationary then ”space” itself will look different for different ”times”.
 
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  • #20
A.T.
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... since this mass crosses no projected space geodesic it is not accelerating in space ...
What does "accelerating in space" mean? The free faller worldline is a geodesic in space-time. If you project it onto space, you get a path that might be curved or straight (radial fall). It has coordiante acceleration (dx/dt != 0) but no proper acceleration (what an accelerometer measures).
 
  • #21
OK I give in. I know you are all right of course. What I also feel is that because people have experienced space and time as separate things since time immemorial it is rather a lot for us to swallow the idea that they are part of spacetime, a thing we had never heard of when I was young. But I'm interested in finding out where this spacetime concept came from and the reasons why it was recognised/invented. I think it was Einstein's idea, based on his mathematics and his desire to understand how God made the universe, so instead of simply quoting the results of his thinking could someone please relate how his reasoning worked: why/how did he invent the spacetime idea?
 
  • #22
Ibix
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I think it was Einstein's idea, based on his mathematics and his desire to understand how God made the universe, so instead of simply quoting the results of his thinking could someone please relate how his reasoning worked: why/how did he invent the spacetime idea?
He didn't. Einstein's 1905 paper justified and explained the Lorentz transforms as the relationship between my space and time coordinates and those of another inertial frame, rather than a mathematical patch to Maxwell's equations as was previously assumed. It was Minkowski who pointed out in 1908 that these transforms could be interpreted as meaning that space and time were part of one 4d structure, christened spacetime.

Einstein then took that idea and ran with it, realising that you could model gravity as curved spacetime. But he did not invent the notion.
 
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  • #23
phinds
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What I also feel is that because people have experienced space and time as separate things since time immemorial ...
And they believed for a VERY long time that the Earth was the center of everything, and they had never heard of quarks, and I could go on and on... Get used to it.
 
  • #24
Ibix
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Get used to it.
Don't think, @woolyhead77, that this stuff came naturally to anyone. The maths of special relativity (published 1905) is implicit in Maxwell's equations (published around 1862). It took forty years to puzzle out, and another ten to expand special relativity to general relativity. And there are still arguments about how best to teach it and it can take years to get your head around it.

But the evidence is compelling. Without GR we can't explain the behaviour of light near masses, nor the exact precession of Mercury' orbit, nor the behaviour of clocks at different heights, nor the redshift of distant galaxies... "Get used to it" is a fairly blunt way of putting it, but an awful lot of "common sense" knowledge about the universe is hilariously wrong outside the limited range of our everyday experience.
 
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  • #25
He didn't. Einstein's 1905 paper justified and explained the Lorentz transforms as the relationship between my space and time coordinates and those of another inertial frame, rather than a mathematical patch to Maxwell's equations as was previously assumed. It was Minkowski who pointed out in 1908 that these transforms could be interpreted as meaning that space and time were part of one 4d structure, christened spacetime.

Einstein then took that idea and ran with it, realising that you could model gravity as curved spacetime. But he did not invent the notion.
quote: "these transforms could be interpreted as meaning that space and time were part of one 4d structure, christened spacetime." Would anyone care to expand on this a bit, please?
 

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