Geometry of Space as a Function of Time: Time Dilation & GR

[Chrisc]

1) If I plotted the time dilation at corresponding, radially symmetric coordinates surrounding a large mass,
would it result in a geometry of space as a function of time?

2) I have heard GR defines the geodesic of a mass moving in the metric of another mass but as it is a geometry
of masses in motion, it does not offer a reason for two masses at rest to begin to move?

 

[Ich]

1.) No, the geometry is static. Such a plot is too abstract (i.e. coordinate dependent) to be more than a visualisation of a certain function. To illustrate the geometry, one usually embeds the equatorial plane in 3D-space. This gives at least the correct distances for slow-speed observers.
2.) The approriate concept is four velocity, which has always norm -1 (robphy:1), and is being tilted when it is transported into the future. In a sense that means that every object is moving already and just changes direction such that part of the movement becomes spatial. Sort of.

 

[Mentz114]

2) I have heard GR defines the geodesic of a mass moving in the metric of another mass but as it is a geometry of masses in motion, it does not offer a reason for two masses at rest to begin to move?

There will be a 'reason' to move. Think of a nearly massless particle released at nearly infinity in a Schwarzschild space-time. The time evolution of the GR equations shows it accelerating towards the source. Is this what you mean ?

M

 

[MeJennifer]

1) it does not offer a reason for two masses at rest to begin to move?

Show me any valid solution in GR where two masses are completely rest at one time and begin to more at another time.

 

[JesseM]

Show me any valid solution in GR where two masses are completely rest at one time and begin to more at another time.

"At rest" depends on your choice of coordinate system, no? And in GR all coordinate systems are equally valid, no matter how weird and arbitrary they are, thanks to diffeomorphism invariance (see this article); so, you could construct a coordinate system where two nearby gravitating objects have a constant coordinate position for some amount and then begin to move towards one another. And even if you pick a situation like a test particle moving under the influence of a large spherical mass in Schwarzschild coordinates, it's not clear that Chrisc was talking about masses at rest for a finite time, you could certainly choose initial conditions where the test particle was instantaneously at rest relative to the spherical mass before beginning to fall at later times.

 

[Chrisc]

I'm not sure there is a consensus on question 1.
Ich, are you saying that time alone is not enough to describe a geometry of space-time?
Or are you saying it is but it is impossible to define the time of each coordinate in any physically
meaningful way from anyone frame?
Or perhaps you're saying something completely different and I just didn't understand?

Mentz114, your answer leaves me asking the same question. Is the "time evolution" as you put it,
an expectation of motion since time does not stand still, or is there an identifiable causal relationship
between the energy of each mass that can be said to act on the other?
My problem is I cannot find any causal relationship in any explanation I've found.
The best explanation of Einstein's equation I've found was John Baez's on his site.
It was very clear on the geometry but left me with the same question.

From Ich's answer to 2, it would seem (in layman's terms) the energy of mass is expressed as constant temporal or spatial displacement, but within the field of another mass temporal gives way to spatial - i.e. motion?

MeJennifer, I cannot solve Einstein's equation. I cannot provide you with an example of two masses completely at rest wrt each other unless a second force hold's them against gravitation. I am not sure I want to go down that road yet, as I don't think I understand enough about the "action" of gravitation in the first place.

JesseM, you lost me.

 

[JesseM]

JesseM, you lost me.

Well, I was making two points: 1) that you can use absolutely any coordinate system in GR, so for any pair of objects you can always find one where they are at rest in that system (i.e. their coordinate position is not changing over time), and 2) even if you use some more "standard" coordinate system like Schwarzschild coordinates, you may find that two objects can be at rest relative to one another for an instant but then begin moving, like if you toss a ball in the air and it is instantaneously at rest relative to the ground at the top of its arc, after which it begins to fall down again.

 

[Dale]

2) I have heard GR defines the geodesic of a mass moving in the metric of another mass but as it is a geometry
of masses in motion, it does not offer a reason for two masses at rest to begin to move?

Don't forget that things like "curvature" are curvature of spacetime, not just space. Even if an object is not moving through space it is "moving" through time.

Of course, this is a rather poor way of describing an object's worldline in spacetime. It is better to think in terms of geometry. If two masses are "at rest" that simply means that their worldlines are parallel. If two worldlines are initially parallel and elsewhere the distance between them changes then one of two things must have happened:

1) one of the lines bent
2) the space itself is not flat

The first case is what happens to the worldline of a mass that experiences a real force, the second case is what happens to the worldline of a mass that begins gravitating towards another mass.

 

[Chrisc]

I think this is making sense, if I put what both of you(JesseM, DaleSpam) said together.
How does this sound?
A pair of coordinates may be at rest in the classical sense, (static with respect to any classical measurement of spatial separation).
But as massive particles they must be under force(a frame permitted by GR) to counter the gravitation between them.
Being under force, they are actually in motion with respect to the gravitational field between them, which is to say,
their world lines are bent with respect to world lines of the same masses not under force, or conversely their world lines are parallel when
under gravitation they should be curved, which relative to the field is straight.
i.e. a geodesic is a straight line in space-time even when that space-time is curved wrt flat space-time.

 

[Dale]

I think you are getting the idea.

However, I should warn you. Using the word "parallel" in curved spaces is somewhat dangerous and likely to get you metaphorically slapped by people who actually know their differential geometry. I don't really know differential geometry very well, so I am comfortable with it even though I know it is sloppy. So just a warning when talking with others.

 

[Chrisc]

Thank you DaleSpam
I think I know what you mean.
I will parallel transport my head beyond the reach of any metaphorical slaps.