Why does bent space set objects in motion?

[Rtenhoor]

Two stationary masses will attract each other. That is, they start moving towards each other. For example, a stationary apple will fall to earth.

I can see bent space affecting the trajectory of a moving object.

...but how does bent space explain that stationary objects start moving?

 

[Mentz114]

Two stationary masses will attract each other. That is, they start moving towards each other. For example, a stationary apple will fall to earth.

I can see bent space affecting the trajectory of a moving object.

...but how does bent space explain that stationary objects start moving?

One explalnation is that everything is moving in time as suggested by the 4-velocity. In the presence of space-time curvature some of the time movement becomes spatial movement. I don't think an explanation is necessary so I won't try to defend this idea.

[Edit]
It turns out that I can defend it because the generators of motion in GR are the Ricci rotation coefficients ${A^\mu}_{\alpha\nu}$ so that $\delta x^\mu={A^\mu}_{\alpha\nu}\delta x^\nu$. For Newton's apple we can write $\delta r = {A^r}_{t\ t}\ \delta t$ and since $\delta t$ is never zero, motion is inevitable provided ${A^r}_{t\ t}$ is not zero. For the Schwarzschild vacuum the value in the apple coords is $-\frac{M}{{r}^{\frac{3}{2}}\,\sqrt{r-2\,M}}$

 

[pervect]

Two stationary masses will attract each other. That is, they start moving towards each other. For example, a stationary apple will fall to earth.

I can see bent space affecting the trajectory of a moving object.

...but how does bent space explain that stationary objects start moving?

General relativity talks about bent space-time and not just bent space. With the proper mathematical techniques (which I won't go into unless requested, and it's an A-level topic), bent space-time can be decomposed into an "bent space" and "bent time". Then bent space only affects the trajectories of moving objects, but "bent time" affects any objects that ages. And all objects age.

 

[PeroK]

Two stationary masses will attract each other. That is, they start moving towards each other. For example, a stationary apple will fall to earth.

I can see bent space affecting the trajectory of a moving object.

...but how does bent space explain that stationary objects start moving?

Check this thread here, for example

https://www.physicsforums.com/threads/how-exactly-does-gravity-work.905157/page-2#post-5930579

 

[PeroK]

I just realized it's a long thread. Here's a key bit:

In Newtonian physics things move according to the forces acting on them. But, a french mathematician called Lagrange reworked Newtonian mechanics into a formulation where things move in order to minimise a quantity called the Lagrangian. Lagrange's formulation of mechanics can be shown to be entirely equivalent to Newton's. But, the underlying reason for motion is somewhat different.

When you move to the theory of General Relativity (GR), there are no forces, so you cannot apply Newton's laws. But, you can apply the Lagrangian principle that nature acts in order to minimise or maximise certain quantities. In GR the quantity being maximised is the "proper" time that a particle experience. This, therefore, is the defining law of GR in respect of the paths that particles take.

In the special case of flat spacetime a particle maximises its proper time by, you guessed it, remaining at rest or moving with constant velocity (in a straight line).

And in the curved spacetime around the Earth, an apple cannot maximise its proper time by staying stationary. The Lagrangian principle compels it to start moving, just as a force would.

 

[sweet springs]

We need another force in order to keep the body stationary in a gravitational field.
I am on the chair now. The chair pushes me up so that I do not fall onto the floor.

Say a fly in baseball game goes up the sky , loses speed and stops at its top height. The ball and its gravitation partner, the Earth seem to stationary state at that moment. But they are not, the ball went up there and will go down. The ball keeps "free falling" even when it was going up.

So in short, any thing does free fall in gravitational field if other forces do not work on it.

 

[Orodruin]

But, a french mathematician called Lagrange reworked Newtonian mechanics into a formulation where things move in order to minimise a quantity called the Lagrangian.

Just to point out that this passage is not entirely accurate. What is done in Lagrangian mechanics is to find the stationary paths (not necessarily minimising paths) of the action (not the Lagrangian), which is the time integral of the Lagrangian.

 

[A.T.]

...but how does bent space explain that stationary objects start moving?

It doesn't. Space-time does.

 

[Ibix]

Another way to look at it is to say that gravity is a modification of Newton's first law. Instead of things following straight lines in space unless acted on by a force they follow geodesics (which are the generalisation of straight lines to non-Euclidean spaces) in spacetime unless acted on by a force. And the projection of a geodesic in spacetime onto space (whatever you precisely mean by that) is generally a curve, as illustrated by A.T.'s animation.

This is an informal view of Lagrange's action-extremisation approach mentioned upthread.

 

[Rtenhoor]

OK, so a quick follow up question then: if total momentum in 4-dim spaceTime stays the same (which I assume it should), then it follows that the 3-dim movement which has started must make the movement in the time-dimension slower.

 

[Mentz114]

OK, so a quick follow up question then: if total momentum in 4-dim spaceTime stays the same (which I assume it should), then it follows that the 3-dim movement which has started must make the movement in the time-dimension slower.

Please fix your post (use the 'Edit' option) so your text is not included in the quote delimiters.

Yes, the body will have the velocity $u^\mu= \frac{r}{2\,M-r} \vec{\partial_t} - \frac{\sqrt{2}\,\sqrt{M}}{\sqrt{r}} \vec{\partial_r}$

 

[Ibix]

OK, so a quick follow up question then: if total momentum in 4-dim spaceTime stays the same (which I assume it should), then it follows that the 3-dim movement which has started must make the movement in the time-dimension slower.

I don't think there's a unique answer to this in curved spacetime. There isn't a general way to compare velocities not at the same event.

I can think of two ways to address this. One is to parallel transport the four momentum along the object's path, in which case the answer is that the momentum doesn't change in any sense. That's what free-fall means, after all. No forces.

The other way is to compare the four momentum to that of a sequence of hovering observers, which is what I think Mentz is doing. In that case the increasing three-monentum leads to an increase in the time component too, since this is a Lorentzian space not a Euclidean one. So the falling observer's clock ticks slower and slower (i.e. for a given interval of their proper time, more and more coordinate time passes) compared to each hovering observer they pass.

So no, or yes-or-no depending on how I answer your question and what you mean by moving slower in the time dimension.

 

[PeterDonis]

if total momentum in 4-dim spaceTime stays the same (which I assume it should)

No, it doesn't. In a general curved spacetime there are no conserved quantities of a geodesic motion. In stationary spacetimes, there is a conserved quantity called "energy at infinity", but it is not the same as "total momentum" or anything like it.

The 4-momentum of an object has a norm equal to its rest mass in the object's instantaneous local inertial frame, but that tells you nothing useful if you're looking at anything beyond a single instantaneous local inertial frame.

 

[Mentz114]

OK, so a quick follow up question then: if total momentum in 4-dim spaceTime stays the same (which I assume it should), then it follows that the 3-dim movement which has started must make the movement in the time-dimension slower.

From the Schwarzschild metric we can write $\dot{\tau}=\sqrt{g_{00}-g_{11}\dot{r}^2}$ so you can see that $\dot{\tau}$ and $\dot{r}$ are inversely related - but this is not conservation. I don't know how general that expression is because it could become imaginary in some metrics.

I don't think there's a unique answer to this in curved spacetime. There isn't a general way to compare velocities not at the same event.
[..]
So no, or yes-or-no depending on how I answer your question and what you mean by moving slower in the time dimension.

Thanks, that has clarified it for me and I hope for the OP.

No, it doesn't. In a general curved spacetime there are no conserved quantities of a geodesic motion. In stationary spacetimes, there is a conserved quantity called "energy at infinity", but it is not the same as "total momentum" or anything like it.

The 4-momentum of an object has a norm equal to its rest mass in the object's instantaneous local inertial frame, but that tells you nothing useful if you're looking at anything beyond a single instantaneous local inertial frame.

Agreed. This is kinematics and the dynamical definition of conservation has to be amended.
Do I remember correctly that geodesic worldlines are parallel to the time-like Killing vector field ?

 

[p1l0t]

Wait so I'm flying through space AND time (or spacetime) right now just sitting here in my chair!?

 

[Ibix]

Wait so I'm flying through space AND time (or spacetime) right now just sitting here in my chair!?

It's not the most precise way of putting it, but yes. Although you can always define yourself as at rest, so not moving through space. No such luck with time, though.

 

[PeterDonis]

Do I remember correctly that geodesic worldlines are parallel to the time-like Killing vector field ?

No. "Stationary" worldlines are integral curves of the timelike KVF. But those worldlines are not geodesics. For example, the worldline of an object sitting on the Earth's surface is an integral curve of the (approximate) timelike KVF of the spacetime in the Earth's vicinity. But it's certainly not a geodesic.

 

[Rtenhoor]

Thinking about this some more, I think maybe the point I missed is that the time dimension is bent by the presence of mass as well.
If it wasn't, two stationary objects would stay stationary, even if the three space dimensions were warped somehow.

 

[Mentz114]

Thinking about this some more, I think maybe the point I missed is that the time dimension is bent by the presence of mass as well.
If it wasn't, two stationary objects would stay stationary, even if the three space dimensions were warped somehow.

This was stated in post#3 and again in subsequent posts.

 

[Rtenhoor]

Mentz114: Right, sorry...

 

[Mentz114]

Mentz114: Right, sorry...

I did not mean to chide you - only to emphasize that you are on the right track. These things are new and take time to sink in.

 

[sergiokapone]

It doesn't. Space-time does.

Does anybody know, how to obtain the transformation matrix from flat grid to curved $(t,y) \to (t',y')$ showed in video?

 

[A.T.]

Does anybody know, how to obtain the transformation matrix from flat grid to curved $(t,y) \to (t',y')$ showed in video?

That's just a conceptual illustration, which is compressed to make the effect more clear. If you are interested in a quantitative diagram, see Chapter 6 of this thesis:

http://www.relativitet.se/Webtheses/lic.pdf

 

[sergiokapone]

That's just a conceptual illustration, which is compressed to make the effect more clear. If you are interested in a quantitative diagram, see Chapter 6 of this thesis:

http://www.relativitet.se/Webtheses/lic.pdf

Thank you. I will try to visualize more quantitative with a mathematical soft. I had tried this with LaTeX, but transformations was wrong, I think

 

[Orodruin]

Locally, it is just the Rindler coordinates discussed in this recent Insight.