What initiates gravitational movement?

[nytmr24]

TL;DR Summary

In GR, if the force of gravity is fictitious, what causes acceleration?

I understand that under the GTR space and time are curved and movement takes place along that curved surface. Setting aside for the moment the additional consideration that the space curvature is in three dimensions, I do not understand the impetus for acceleration. If an object is just sitting alone in curved space with no movement with respect to any other object, what could cause it to move, if there are no forces involved? I also understand that the object is not fixed in time, so it is moving with respect to the time coordinate axis. Does its movement along its time line cause its movement in the spatial dimensions?
Can anyone help me understand this?

 

[Grasshopper]

I would imagine it’s simply that objects always move along geodesics. Just like in Newtonian physics, objects will move in a straight line through space with no force acting on them.

But don’t take my word for it, I know nothing.

 

[PeterDonis]

I would imagine it’s simply that objects always move along geodesics.

Basically, yes.

More precisely, what you are thinking of as "moving" vs. "not moving" is not correct. "Moving" is frame-dependent; there is no invariant sense in which any object is "moving" or "not moving", and it makes no sense to ask whether a particular object is moving or not moving.

What it does make sense to ask is what trajectory through spacetime (not "space") an object takes, and why. And the answer is that, in the absence of any forces (gravity is not a force in GR) acting on an object, its trajectory through spacetime will be a geodesic of spacetime. That is an invariant statement and doesn't depend on any choice of frame. And it is all the "cause" that GR requires to explain the trajectories of objects that don't have any forces acting on them.

 

[A.T.]

Does its movement along its time line cause its movement in the spatial dimensions?

Yes. See video below:

 

[A.T.]

I would imagine it’s simply that objects always move along geodesics. Just like in Newtonian physics, objects will move in a straight line through space with no force acting on them.

The following is true in both, Newtonian Physics and General Relativity:

Objects that have no net force on them progress along geodesic wordlines in spacetime.

The difference between the two is that in Newtonian Physics gravity is a force that counts towards that net force, while in General Relativity it isn't.

 

[PeterDonis]

The following is true in both, Newtonian Physics and General Relativity:

Objects that have no net force on them progress along geodesic wordlines in spacetime.

The difference between the two is that in Newtonian Physics gravity is a force that counts towards that net force, while in General Relativity it isn't.

This isn't quite correct. In Newtonian Physics there is no concept of geodesics in spacetime, only in space. (This is true even in the geometric Cartan formulation of Newtonian physics, because there is no spacetime metric in that formulation.) In Newtonian physics, objects with no force acting on them (including gravity as a force, as you say) have geodesic trajectories in space, where "space" means the Euclidean 3-space that is "absolute space" in Newtonian theory.

 

[A.T.]

In Newtonian Physics there is no concept of geodesics in spacetime, only in space.

In Newtonian Physics geodesics in spacetime correspond to straight lines in a space-time diagram.

 

[vanhees71]

But there is no connection in Newtonian spacetime, at least not a physically relevant one, and thus you cannot define geodesics in Newtonian spacetime. Imho you don't necessarily need a (pseudo-)metric to define geodesics, but some connection is sufficient. Then you define a geodesic as a curve in the manifold with connection along which the tangent vectors are parallel transported, i.e., as the "straightest possible lines" in the sense of the connection.

 

[A.T.]

But there is no connection in Newtonian spacetime, at least not a physically relevant one, and thus you cannot define geodesics in Newtonian spacetime.

A straight line is just a special case of a geodesic.

 

[vanhees71]

AFAIK in Newtonian spacetime there is no (physically natural) connection. So you can neither define a straight line nor a geodesic to begin with.

 

[PeterDonis]

In Newtonian Physics geodesics in spacetime correspond to straight lines in a space-time diagram.

No, they don't, because, as I've said, there is no meaningful notion of "geodesics in spacetime" in Newtonian Physics. Just waving your hands and saying "well, it's a straight line in a spacetime diagram" is not enough. You have to have a spacetime metric and a metric-compatible connection for the notion of "geodesics in spacetime" to be meaningful. You don't have a spacetime metric in Newtonian Physics.

A straight line is just a special case of a geodesic.

This is only true if there is a meaningful notion of "geodesic" for the manifold in question to begin with.

 

[A.T.]

In Newtonian Physics geodesics in spacetime correspond to straight lines in a space-time diagram.

No, they don't, because, as I've said, there is no meaningful notion of "geodesics in spacetime" in Newtonian Physics. Just waving your hands and saying "well, it's a straight line in a spacetime diagram" is not enough.

I wrote "correspond to" not "it's a".

 

[PeterDonis]

I wrote "correspond to" not "it's a".

Doesn't matter. Something that isn't even well-defined in the first place can't "correspond to" anything.

 

[pervect]

I don't think it's standard to say that Newtonian mechanics has a connection in space-time, and I agree with the comments that in Newton -Cartan theory, there is no such connection - at least as much as I remember about Newton-Cartan theory.

However, it also seems to me that an inertial frame of reference in Newtonian mecahnics defines a connection. I.e. if we have cartesian coordinates t,x,y,z in some inertial frame, $\partial / \partial t, \,\partial / \partial x, \, \partial / \partial y, \, \partial / \partial z$ can be regarded as pointing in the "same direction" in space-time at any event in the inertial frame, creating a map from the tangent space at one event in space-time to another event, all we need to create a connection.

Gravity in this viewpoint is regarded as a force, an instantaneous action at a distance, rather than geometrically as it is in Newton-Cartan theory.

 

[PeterDonis]

it also seems to me that an inertial frame of reference in Newtonian mecahnics defines a connection. I.e. if we have cartesian coordinates t,x,y,z in some inertial frame, $\partial / \partial t, \,\partial / \partial x, \, \partial / \partial y, \, \partial / \partial z$ can be regarded as pointing in the "same direction" in space-time at any event in the inertial frame, creating a map from the tangent space at one event in space-time to another event, all we need to create a connection.

This obviously works for the spatial partial derivatives (and in fact is basically what defines geodesics of "space" as I was using that term in an earlier post), because, roughly speaking, a Galilean transformation doesn't change which direction in spacetime they point--surfaces of constant time are invariant under Galilean transformations.

However, I don't think it works for $\partial / \partial t$ because a Galilean transformation does change which direction in spacetime that vector points, and that seems to me to be inconsistent with the fact that the connection defined as you describe should be the same in every inertial frame, i.e., it should be invariant under Galilean transformations.

 

[pervect]

However, I don't think it works for $\partial / \partial t$ because a Galilean transformation does change which direction in spacetime that vector points, and that seems to me to be inconsistent with the fact that the connection defined as you describe should be the same in every inertial frame, i.e., it should be invariant under Galilean transformations.

In the absolute time of Newtonian theory, how does the Gallilean transformation t'=t change the direction of time?

 

[PeterDonis]

In the absolute time of Newtonian theory, how does the Gallilean transformation t'=t change the direction of time?

It changes the "direction in spacetime" that the tangent vector to the worldline of an observer "at rest" points, which is what $\partial / \partial t$ is--more precisely, what it is if you want to use it as a basis vector in order to define a connection. (When you try to fully unpack what this implies, you realize why there isn't any spacetime metric in Newtonian gravity.)

 

[DrGreg]

In the absolute time of Newtonian theory, how does the Gallilean transformation t'=t change the direction of time?

Newhouse's second fundamental confusion of calculus: $$t' = t$$ does not imply $$\frac{\partial}{\partial t'} = \frac{\partial}{\partial t}$$

 

[hutchphd]

Meanwhile, the OP probably would like a simpler answer?

Summary:: In GR, if the force of gravity is fictitious, what causes acceleration?

If an object is just sitting alone in curved space with no movement with respect to any other object, what could cause it to move, if there are no forces involved? I also understand that the object is not fixed in time, so it is moving with respect to the time coordinate axis. Does its movement along its time line cause its movement in the spatial dimensions?

If there are no forces involved there will be no change in relative motion. But if I am standing on Earth and drop an apple, only the apple suddenly sees no forces: my feet are still securely pushed by electromagnetic forces from the dirt and so I continue to be accelerated while the apple is stops accelerating. We begin to move apart.

 

[A.T.]

Doesn't matter. Something that isn't even well-defined in the first place can't "correspond to" anything.

In General Relativity:
If the net force (excluding gravity) on a body is zero, its path in spacetime is a geodesic.

In Newtonian Physics:
If the net force (including gravity) on a body is zero, its path in spacetime is a straight line.

 

[vanhees71]

However, I don't think it works for $\partial / \partial t$ because a Galilean transformation does change which direction in spacetime that vector points, and that seems to me to be inconsistent with the fact that the connection defined as you describe should be the same in every inertial frame, i.e., it should be invariant under Galilean transformations.

In which way? The most general proper orthochronous Galilei transformation should be, expressed in Cartesian coordinates of configuration-space coordinate $\vec{x}$ and time $t$
$$t'=t+a, \quad \vec{x}'=\hat{D} \vec{x} -\vec{v} t + \vec{b},$$
where $\hat{D} \in \mathrm{SO}(3)$, $a \in \mathbb{R}$, $\vec{v},\vec{b} \in \mathrm{R}^3$.

In Newtonian physics time and space are absolute, i.e., there's no change of direction of a "space-time vector". Newtonian spacetime is more a fibre bundle than some kind of 4D affine space like in SRT.

 

[PeterDonis]

In which way?

Pick a particular inertial frame. A worldline at rest in this frame will have equation $x(t) = x_0$. A worldline at rest in a different frame will, in our chosen frame, have equation $x(t) = x_0 + v t$, where $v$ is the relative velocity between the frames. The tangent vectors to these two different worldlines point in different directions in spacetime. And these tangent vectors correspond to $\partial / \partial t$ in the coordinates corresponding to the two different frames.

In Newtonian physics time and space are absolute, i.e., there's no change of direction of a "space-time vector".

If you want to say there is no such thing as a "space-time vector" to begin with, then of course something that isn't well-defined can't change direction. I have been using an implicit notion of "space-time vector" for Newtonian physics that appears to me to make sense, but I have not gone into the technical details to verify that it actually is well-defined mathematically.

If you agree that there is a well-defined notion of "space-time vector" in Newtonian physics, then I don't see how any such notion would say that the two different tangent vectors I described above would not point in different directions in spacetime.

 

[A.T.]

AFAIK in Newtonian spacetime there is no (physically natural) connection. So you can neither define a straight line nor a geodesic to begin with.

A straight line in Newtonian spacetime means to me, that the 2nd derivatives of position with respect to time are zero.

 

[vanhees71]

That's a straight line in configuration space, not spacetime.

 

[cmb]

Basically, yes.

More precisely, what you are thinking of as "moving" vs. "not moving" is not correct. "Moving" is frame-dependent; there is no invariant sense in which any object is "moving" or "not moving", and it makes no sense to ask whether a particular object is moving or not moving.

What it does make sense to ask is what trajectory through spacetime (not "space") an object takes, and why. And the answer is that, in the absence of any forces (gravity is not a force in GR) acting on an object, its trajectory through spacetime will be a geodesic of spacetime. That is an invariant statement and doesn't depend on any choice of frame. And it is all the "cause" that GR requires to explain the trajectories of objects that don't have any forces acting on them.

This may just be a conflict between intuition and equations, but surely the point between me and the chair I am sitting on is 'a point' traveling along the same geodesic, so why is the chair pushing against me one way ('up') and I am pushing against it the other ('down'), if there is only one point of contact between us which can only be following one path through spacetime as it is just one point?

 

[PeterDonis]

surely the point between me and the chair I am sitting on is 'a point' traveling along the same geodesic

If you are sitting on a chair, you are not traveling on a geodesic. You can tell that by the fact that you feel weight. If you were traveling on a geodesic (like, for example, the astronauts aboard the International Space Station), you would be weightless.

why is the chair pushing against me one way ('up') and I am pushing against it the other ('down')

Newton's Third Law.

 

[A.T.]

This may just be a conflict between intuition and equations, but surely the point between me and the chair I am sitting on is 'a point' traveling along the same geodesic,

Unless you and your chair are in free fall, they are not following a geodesic in spacetime.

... so why is the chair pushing against me one way ('up') and I am pushing against it the other ('down'), if there is only one point of contact between us which can only be following one path through spacetime as it is just one point?

The net force determines the path in space time. not just one of the forces acting. Your bottom layer has also forces from the rest of your body acting on it. The top layer of the chair has also forces from the rest of chair acting on it. The net forces on both are such that they follow similar nearby paths in spacetime.

 

[cmb]

If you are sitting on a chair, you are not traveling on a geodesic. You can tell that by the fact that you feel weight. If you were traveling on a geodesic (like, for example, the astronauts aboard the International Space Station), you would be weightless.

Oh, I see, I didn't realize 'that term' mean that following a geodesic in space time means you're weightless (I think that's what you just said?).

Newton's Third Law.

Well, yeah, heh!

Not sure Newton was quite up to speed on space time geodesics though? ;)

 

[PeterDonis]

following a geodesic in space time means you're weightless (I think that's what you just said?)

Yes.

Not sure Newton was quite up to speed on space time geodesics though?

Newton's Third Law still applies in General Relativity. The only difference is that in GR gravity is not a force, so it makes no sense to apply Newton's Third Law to it. But the force between you and the chair is not gravity, so Newton's Third Law applies there.

 

[A.T.]

Not sure Newton was quite up to speed on space time geodesics though? ;)

The net force from Newton's 2nd is still determines the spacetime path in GR, except that gravity is not included in that net force.

While you are pushing down on the chair, the ground is pushing up on the chair even more. So both you and the chair have net upwards forces on them, and thus upwards proper accelerations of 1g, as the accelerometer in you phone will confirm. Thus, neither you nor your chair are following a geodesic in spacetime.