How Does Spacetime Manipulation Affect the Flow of Time?

[Yuripe]

If you look at universe as 3+1 dimensional space, all you can see are stationary objects (particles, matter) distributed along space and time.
To each object in this spacetime you can assign a vector V which describes its dynamics. This dynamics is to be understood as rate of transition from one point in spacetime to the next V(dx,dy,dz,dt) another words - speed.

The speed of light would be the maximum observable length of V.
The difference of lenghts for each pair of Vs would represent relative speed between them.
The information about length of a V is distributed no faster then the speed of light from the point of origin of a V to any given point in spacetime.
The values of dx,dy,dz and dt also depend on local "density" of spacetime.

Time dilation would be caused by

• gravity - local spacetime compression (acceleration field) influencing value of dt among others
• the limited speed of information flow from the object to the observer (distance and difference in lengths of Vs).

All above doesn't come even close to describing why dt is always non-zero and what determines its value. We just simply call it timelike dimension.

I'll rephrase my original question and ask this:
Could the value of dt or at least its always non-zero property be attributed in some way to the expanding spacetime?

 

[Dale]

If you look at universe as 3+1 dimensional space, all you can see are stationary objects (particles, matter) distributed along space and time.
To each object in this spacetime you can assign a vector V which describes its dynamics. This dynamics is to be understood as rate of transition from one point in spacetime to the next V(dx,dy,dz,dt) another words - speed.

I think you are talking about the tangent vector which is also known as the four-velocity. If so, you are definitely on the right track, IMO.

The speed of light would be the maximum observable length of V.

The length of the four-velocity of light is 0, but the length of the four-velocity for any massive particle is c.

The difference of lenghts for each pair of Vs would represent relative speed between them.

Actually, the relative speed is related geometrically to the angle between them.

The information about length of a V is distributed no faster then the speed of light from the point of origin of a V to any given point in spacetime.
The values of dx,dy,dz and dt also depend on local "density" of spacetime.

Time dilation would be caused by

• gravity - local spacetime compression (acceleration field) influencing value of dt among others
• the limited speed of information flow from the object to the observer (distance and difference in lengths of Vs).

The important thing about c is that it is frame invariant. That is what causes time dilation. Suppose that we were in an opaque medium and signals had to be sent acoustically. Information then could only travel at the speed of sound, but since that is not frame invariant it would not cause time dilation.

All above doesn't come even close to describing why dt is always non-zero and what determines its value. We just simply call it timelike dimension.

I'll rephrase my original question and ask this:
Could the value of dt or at least its always non-zero property be attributed in some way to the expanding spacetime?

I don't think so. After all, what you describe would be just as relevant in a static Minkowski spacetime as in an expanding FLRW spacetime.

 

[Rasalhague]

Just a footnote to DaleSpam's post: be sure to distinguish between 4-velocity, the directional derivative with respect to arc length, i.e. proper time, along a timelike curve (timelike worldline), and 3-velocity, what we normally call velocity outside of the context of relativity: the derivative of position with respect to coordinate time. 3-velocity is coordinate dependent, whereas 4-velocity is not. I've also seen 3-velocity called "relative velocity". The magnitude of the 4-velocity, considered as a tangent vector, along a timelike curve is c or -c depending which sign convention [ http://en.wikipedia.org/wiki/Sign_convention ] you use, (1,-1,-1,-1) or (-1,1,1,1). Often people use what are called "geometric units" of time and length where c = 1, such as years and light years, in which case the magnitude is 1 or -1, depending on sign convention.

Light has the maximum 3-velocity, but any lightlike 4-vector (a vector parallel to a lightlike worldline) has magnitude zero. But zero isn't necessarily the minimum magnitude of a 4-vector. There are a variety of definitions of magnitude used, according to some of which there can be negative as well as positive magnitudes depending on whether the 4-vector is timelike or spacelike. According to other common definitions, timelike and spacelike 4-vectors are distinguished by real versus imaginary magnitudes, again depending on sign convention. It can be bewildering sorting through all these conventions, but then what doesn't kill you makes you stronger: I guess it helps to think about what's essential to the theory and what's just an arbitrary convention.

 

[Yuripe]

The important thing about c is that it is frame invariant. That is what causes time dilation.

How would you determine the value of your own speed in expanding spacetime where everything else in it generally moves away from each other?

Suppose that we were in an opaque medium and signals had to be sent acoustically. Information then could only travel at the speed of sound, but since that is not frame invariant it would not cause time dilation.

In this example I think speed of sound isn't frame invariant because you observe it with the speed of light. If you would reduce the speed of light so it equals the speed of sound then sound waves would appear frame invariant.

 

[Dale]

How would you determine the value of your own speed in expanding spacetime where everything else in it generally moves away from each other?

There is no meaning to this. Only relative speeds have meaning, and in a curved spacetime only if the two objects are near enough to each other to neglect the curvature.

In this example I think speed of sound isn't frame invariant because you observe it with the speed of light. If you would reduce the speed of light so it equals the speed of sound then sound waves would appear frame invariant.

You can easily come up with acoustic/mechanical experiments that don't involve anything related to the speed of light.