# GR gravity path confusion

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Andreas C
Ok. I get that objects seem to fall because of curved spacetime, when they're actually just moving in straight paths. I get the example of ants walking in straight lines on the surface of a sphere, thinking that something attracts them to each other. What I don't get is how the "ants" are "attracted" to each other without walking. What that means is, how is a stationary object attracted to another stationary object? I don't get this.

Gold Member
Ok. I get that objects seem to fall because of curved spacetime, when they're actually just moving in straight paths. I get the example of ants walking in straight lines on the surface of a sphere, thinking that something attracts them to each other. What I don't get is how the "ants" are "attracted" to each other without walking. What that means is, how is a stationary object attracted to another stationary object? I don't get this.
You're thinking of curved space only. In curved spacetime, stationary objects are effectively moving along the time axis at speed c, so the curvature of free fall paths as the object moves through time accelerates it even if it isn't moving in space.

Andreas C
Gold Member
2021 Award
Ok. I get that objects seem to fall because of curved spacetime, when they're actually just moving in straight paths. I get the example of ants walking in straight lines on the surface of a sphere, thinking that something attracts them to each other. What I don't get is how the "ants" are "attracted" to each other without walking. What that means is, how is a stationary object attracted to another stationary object? I don't get this.
Start with the first scenario you posited. An object "seems to fall because of curved spacetime". Now expand it to encompass the "falling" object and the object towards which is it falling. Now you have exactly the scenario in your question. Two objects "attract" each other because of the space-time geometry between them. If they are stationary relative to each other, that means that there is a force involved that is keeping them apart. In your first part, you said they are moving towards each other with means no, or small, other force. In the part where you say they are stationary you have (without realizing it) posited another force. A good example of this is you standing on the surface of the Earth. the force keeping you from falling into the Earth towards the center is the electrostatic repulsion of your feet with the ground. If you jumped out of an airplane, there would be no such force so the space-time geometry would cause you to move towards the center of the Earth.

Andreas C
You're thinking of curved space only. In curved spacetime, stationary objects are effectively moving along the time axis at speed c, so the curvature of free fall paths as the object moves through time accelerates it even if it isn't moving in space.

Ah ok, I think I get it now. So it moves in time, but it doesn't in space.

Andreas C
Start with the first scenario you posited. An object "seems to fall because of curved spacetime". Now expand it to encompass the "falling" object and the object towards which is it falling. Now you have exactly the scenario in your question. Two objects "attract" each other because of the space-time geometry between them. If they are stationary relative to each other, that means that there is a force involved that is keeping them apart. In your first part, you said they are moving towards each other with means no, or small, other force. In the part where you say they are stationary you have (without realizing it) posited another force. A good example of this is you standing on the surface of the Earth. the force keeping you from falling into the Earth towards the center is the electrostatic repulsion of your feet with the ground. If you jumped out of an airplane, there would be no such force so the space-time geometry would cause you to move towards the center of the Earth.

You don't have to be affected by a force in order not to be moving relative to another object... I think you misunderstood what I said.

Gold Member
2021 Award
You don't have to be affected by a force in order not to be moving relative to another object... I think you misunderstood what I said.
Oh? How do you figure that? Give me an example

Ah ok, I think I get it now. So it moves in time, but it doesn't in space.
Yes, initially. But due to the space-time geometry it deviates from the purely temporal direction, and starts moving in space.

nnunn, Andreas C and Nugatory
m4r35n357
Ok. I get that objects seem to fall because of curved spacetime, when they're actually just moving in straight paths. I get the example of ants walking in straight lines on the surface of a sphere, thinking that something attracts them to each other. What I don't get is how the "ants" are "attracted" to each other without walking. What that means is, how is a stationary object attracted to another stationary object? I don't get this.
You are nearly there, you already have it in terms of space. Two ants at the equator will approach each other as they move towards the "North pole". Now think of the latitude as a time axis, and you have rudimentary "gravitation".

Andreas C and Frimus
Andreas C
Oh? How do you figure that? Give me an example

Why would that be the case? Can't two bodies be stationary in space? Since when? I think you're using the conclusion from what I want explained to explain it to me. Anyway, my question was explained.

Andreas C
You are nearly there, you already have it in terms of space. Two ants at the equator will approach each other as they move towards the "North pole". Now think of the latitude as a time axis, and you have rudimentary "gravitation".

Oh, I think I get it now, I can picture it now in 2 spatial dimensions, I obviously can't in 3 :)

Gold Member
2021 Award
Why would that be the case? Can't two bodies be stationary in space? Since when? I think you're using the conclusion from what I want explained to explain it to me. Anyway, my question was explained.
No, they cannot. Gravity will pull them together unless they are kept apart by some other force. Any other examples? (hint ... there aren't any).

Oh, I think I get it now, I can picture it now in 2 spatial dimensions, I obviously can't in 3 :)
You can visualize a curved 2D space-time, but one of the dimensions needs to be time, so you actually have just 1 spatial dimension (like the straight vertical fall in the video I posted above). Correctly visualizing it for 2 spatial dimestions + time (for example to show orbits as geodesics) is difficult,

Andreas C
No, they cannot. Gravity will pull them together unless they are kept apart by some other force. Any other examples? (hint ... there aren't any).

That's what I am saying. You are using the conclusion coming from what I want explained to explain it.
-How can bodies be "pulled together" due to spacetime geometry by gravity in GR if they are stationary?
-They can't be stationary.
-
Why can't they be stationary?
-Because they are pulled together by gravity.

You probably misunderstood my question.

Gold Member
2021 Award
That's what I am saying. You are using the conclusion coming from what I want explained to explain it.
-How can bodies be "pulled together" due to spacetime geometry by gravity in GR if they are stationary?
-They can't be stationary.
-
Why can't they be stationary?
-Because they are pulled together by gravity.

You probably misunderstood my question.
Must have.

Andreas C
Andreas C
You can visualize a curved 2D space-time, but one of the dimensions needs to be time, so you actually have just 1 spatial dimension (like the straight vertical fall in the video I posted above). Correctly visualizing it for 2 spatial dimestions + time (for example to show orbits as geodesics) is difficult,

Ah, no, I only visualized the concept of 2 particles on a 2-d surface being attracted to each other, nothing more complicated than that! I imagined something like 2 balls inside a sphere. The height of the sphere is time. Ok, it's not accurate, it's just the concept that I wanted to visualize, thanks!

Ah, no, I only visualized the concept of 2 particles on a 2-d surface being attracted to each other, nothing more complicated than that!
That's fine. For two bodies pulled together along a straight line one spatial dimension is enough (the one along that line). The 2nd dimension on your sphere is time.

But what you are visualizing here is how particles of negligible mass are pulled together by the gravity gradient of a bigger mass (tidal gravity), not how masses attract each-other with their own mass.

Staff Emeritus
That's what I am saying. You are using the conclusion coming from what I want explained to explain it.
-How can bodies be "pulled together" due to spacetime geometry by gravity in GR if they are stationary?
-They can't be stationary.
-
Why can't they be stationary?
-Because they are pulled together by gravity.

You probably misunderstood my question.

I would say that the idea of gravity as curvature only works when you think that we reside in 4-dimensional spacetime, rather than 3-dimensional space. And furthermore, every object has a nonzero velocity in the time-direction (we are all moving into the future).

If you think of time as a dimension like space, then how gravity works is perhaps easier to understand. When you say that an object is "stationary", that only means that the spatial components of its velocity are zero, but the time component of its velocity is nonzero: It is moving into the future. Now, add gravity, and that path becomes bent--it starts moving purely in the time direction, but bends to start moving in a spatial direction, as well.

Markus Hanke
How can bodies be "pulled together" due to spacetime geometry by gravity in GR if they are stationary?

Because they age into the future; see post #17 for details.

Andreas C
That's fine. For two bodies pulled together along a straight line one spatial dimension is enough (the one along that line). The 2nd dimension on your sphere is time.

But what you are visualizing here is how particles of negligible mass are pulled together by the gravity gradient of a bigger mass (tidal gravity), not how masses attract each-other with their own mass.

Sure. I just wanted to realize the concept.

Andreas C
I would say that the idea of gravity as curvature only works when you think that we reside in 4-dimensional spacetime, rather than 3-dimensional space. And furthermore, every object has a nonzero velocity in the time-direction (we are all moving into the future).

If you think of time as a dimension like space, then how gravity works is perhaps easier to understand. When you say that an object is "stationary", that only means that the spatial components of its velocity are zero, but the time component of its velocity is nonzero: It is moving into the future. Now, add gravity, and that path becomes bent--it starts moving purely in the time direction, but bends to start moving in a spatial direction, as well.

Yeah, that was explained by Jonathan Scott and A.T.

MeJennifer
In curved spacetime, stationary objects are effectively moving along the time axis at speed c
There is no such thing as a time axis in spacetime.

Time is the distance traveled on a particular world line, each world line has its unique sense of time.

MeJennifer
And furthermore, every object has a nonzero velocity in the time-direction (we are all moving into the future).
That is actually a meaningless statement in GR. World lines extend all the same way but world lines could approach from opposite directions in some models. So which direction is the future?

nnunn
Staff Emeritus
That is actually a meaningless statement in GR. World lines extend all the same way but world lines could approach from opposite directions in some models. So which direction is the future?

I think that there might be exotic solutions of the equations of GR for which there is no consistent way to assign a notion of "future" to timelike paths, but in our universe, there is a unique notion of "the future".

MeJennifer
.. in our universe, there is a unique notion of "the future".
You don't know that!

Staff Emeritus
There is no such thing as a time axis in spacetime.

Time is the distance traveled on a particular world line, each world line has its unique sense of time.

You're right, that there is no such thing as a time axis, but at every point in spacetime, we can pick 4 independent axes such that one of them is timelike and the others are spacelike. And the point is that the spacetime path of any particle will have a nonzero "velocity" with respect to any timelike axis.

Staff Emeritus
You don't know that!

Okay, all the evidence points to a Big Bang cosmology that does have a unique directionality in time. The model could be wrong, of course, but just about any non-tautological statement could be wrong.

nnunn
eltodesukane
Ok. I get that objects seem to fall because of curved spacetime, when they're actually just moving in straight paths. I get the example of ants walking in straight lines on the surface of a sphere, thinking that something attracts them to each other. What I don't get is how the "ants" are "attracted" to each other without walking. What that means is, how is a stationary object attracted to another stationary object? I don't get this.
Remember that 4D spacetime is curved, not 3D space. A stationary object in spacetime is a line, not a point. A stationary object in space is moving into the future.

MeJennifer
And the point is that the spacetime path of any particle will have a nonzero "velocity" with respect to any timelike axis.
As long as the particle has mass.

Mentor
the spacetime path of any particle will have a nonzero "velocity" with respect to any timelike axis.

Which mathematical object are you referring to here? The norm of the tangent vector to the particle's worldline? If so, MeJennifer's statement that this only applies to a massive particle is correct. However, the norm is invariant and you seem to be talking about something that could change from one inertial frame (choice of timelike axis) to another, so I'm not sure exactly what mathematical object you are referring to.

Staff Emeritus
Which mathematical object are you referring to here?

The component of the 4-velocity of the particle in the "time" direction. Mathematically, let $e_\mu$ be any basis for which $e_0$ is timelike and $e_j$ is spacelike, for $j \neq 0$. Let $U^\mu$ be a particle's 4-velocity expressed in this basis. Then $U^0 \neq 0$. I think that's true. If so, it's a statement that is true in any locally Minkowskian frame, but it's not clear how to express it as a statement about invariants.

Gold Member
2021 Award
A classical particle is always on-shell and thus you have
$$g_{\mu \nu} u^{\mu} u^{\nu}=1$$
since
$$u^{\mu}=\frac{1}{m} p^{\mu},$$
where ##p^{\mu}## is the momentum of the particle.

The four-velocity is always time like and ##u^0 >0## by convention, i.e., you take the proper time defining the same causal time direction as the coordinate time.

Staff Emeritus
The four-velocity is always time like and ##u^0 >0## by convention, i.e., you take the proper time defining the same causal time direction as the coordinate time.

$u^0 \neq 0$ is what I was trying to say in post #25.

MeJennifer
A mass less particle does not have a four velocity.

Staff Emeritus
It has a parametrized path $x^\mu(s)$ (although $s$ is not unique), and associated with every parametrized path is a 4-vector whose components are $\frac{d x^\mu}{ds}$. The only difference with a massive particle is that $s$ can't be proper time.