Exploring Time's Flow in Spacetime

[Dale]

We are describing time as the one dimension of spacetime but you can see that it's a little different from the other dimensions. We're not free to stop in time or change direction of our movement in it as we can do in spatial dimensions.
The metric quoted earlier by DaleSpam works regardless of the direction of movement in time dimension and that's fine, but we don't observe such freedom in real world.

The English gets a little difficult here, but in the geometric view of spacetime we are not "moving" through time nor are we "moving" through space. We have some worldline which is fixed and unchanging in spacetime. That worldline may have different tangent vectors at different points along it, but there is not a red dot labeled "you are here" which slides along the worldline.

Sorry if that is confusing, it is a weird concept and the words are rather imprecise. In the geometric view we cannot have dt=0 along our worldline because our worldline is timelike. If it were spacelike then we could not have dx=0. Timelike and spacelike are frame invariant concepts.

 

[Rasalhague]

Coordinate free measurements of distance exist, for instance radar distance and ruler distance.

I don't understand, unless by distance you mean spectime interval (in flat spacetime), or spacetime arc length (more generally), rather than spatial distance. The derivations of the length contraction formula that I've seen in elementary textbooks typically use a radar style measurement, although they may call it a pulse of light. The equations for a Lorentz boost, which show how distance measurements change with one kind of change of coordinates, don't specify "except for distances measured by the following devices". Don't they purport to be true whatever kind of equipment is used, in so far as that equipment is accurate?

 

[Rasalhague]

We are describing time as the one dimension of spacetime but you can see that it's a little different from the other dimensions. We're not free to stop in time or change direction of our movement in it as we can do in spatial dimensions.
The metric quoted earlier by DaleSpam works regardless of the direction of movement in time dimension and that's fine, but we don't observe such freedom in real world.

You're right, the metric doesn't single out a preferred orientation of past and future, and at the level of very simple systems of just a few particles, I gather, there is no preferred orientation. Even at a macroscopic scale, if you look at a simplified model of planets moving under the influence of gravity, "playing the film backwards" doesn't look fundamentally different, except that they'd be moving in the opposite spatial direction. The difference we observe between past and future that makes it possible for us to agree on which is which seems to be something statistical that emerges from complex systems. I find all this mysterious and fascinating, and certainly don't claim to understand it. I keep mentioning thermodynamics because I think some of the answers to your questions may lie there; but haven't got very far with my studies of this, so I can't say much useful about it.

But although the metric doesn't tell you which is the best way to call the future, the fact that this metric isn't changed by circular or hyperbolic rotations of coordinates makes it impossible to reverse time orientation by such means. No combinations of turns or changes of velocity will turn your past into your future and vice-versa, whatever that would mean physically.

Changing direction. We can change direction in space because there's more than one dimension of space. I can take a step north, then a step east. I don't have to stick to the north south line all the time. When an object changes its speed, even if it doesn't change its direction of movement through space, there is something analogous to a change of direction that happens in spacetime: that hyperbolic rotation I mentioned. This is covered in any good introduction to special relativity. It's an interchange of time and space coordinates, analogous to the way a circular rotation interchanges different space coordinates. As there is only one dimension of time in this model, there's never a circular rotation-like interchange of two time coordinates, and that's something that makes time different from space.

The speed of light being a cosmic speed limit, and the related geometric properties of flat spacetime (non-curved spacetime, spacetime without gravity), make it impossible to have a "closed timelike curve" in flat spacetime: a traversible path that forms a loop. As far as I know, it's an open question whether such a thing could exist in curved spacetime. If it did, the article I linked to argues, there would be no natural way of specifying which direction was the future for the whole closed curve. The local future at one point on the curve would inevitably be pointing towards the past from the perspective of another point on the curve. What this would mean in practice, I don't know!

 

[Dale]

Radar distance works as follows: a clock is attached to a radar transciever, a radar pulse is emitted at te and the reflection is received at tr. The proper time tr-te is a frame invariant quantity and the radar distance is c/2 times that proper time. Other observers will not agree that this procedure correctly measures distance, but they will agree on what value the measurement produces.

EDIT: see below, I made a mistake

 

[Dale]

But although the metric doesn't tell you which is the best way to call the future, the fact that this metric isn't changed by circular or hyperbolic rotations of coordinates makes it impossible to reverse time orientation by such means. No combinations of turns or changes of velocity will turn your past into your future and vice-versa, whatever that would mean physically.

That is a really good point that I hadn't seen expressed like that. Thinking geometrically, a surface of constant spacetime interval forms a 4D hyperboloid. Spacelike intervals make hyperboloids of one sheet but timelike intervals make hyperboloids of two sheets. That is the geometric reason that you cannot smoothly transform a future directed timelike interval into a past directed timelike interval.

 

[Passionflower]

But although the metric doesn't tell you which is the best way to call the future, the fact that this metric isn't changed by circular or hyperbolic rotations of coordinates makes it impossible to reverse time orientation by such means. No combinations of turns or changes of velocity will turn your past into your future and vice-versa, whatever that would mean physically.

General relativity does not exclude the possibility of closed timelike curves (CTCs). The theory cannot determine however if such scenarios are physically possible.

 

[Rasalhague]

General relativity does not exclude the possibility of closed timelike curves (CTCs). The theory cannot determine however if such scenarios are physically possible.

Good point, I rearranged my post as I wrote it, and forgot this paragraph now comes before the final one with the proviso about it being only forbidden in flat spacetime, and the possibility of a CTS in curved spacetime.

Radar distance works as follows: a clock is attached to a radar transciever, a radar pulse is emitted at te and the reflection is received at tr. The proper time tr-te is a frame invariant quantity and the radar distance is c/2 times that proper time. Other observers will not agree that this procedure correctly measures distance, but they will agree on what value the measurement produces.

If "other observers" is synonymous with "other coordinate systems", doesn't this make it a coordinate dependent measurement of distance like any other? Well, maybe it's just a matter of words, but if we can adopt a special definition of distance in this instance, why not in any instance, and just define a particular distance measured in a particular way as "invariant". But I suppose that would undermine the distinction between "coordinate dependent" and "invariant", causing us to have to find new words for those concepts, which would be a nuisence...

 

[Phrak]

General relativity does not exclude the possibility of closed timelike curves (CTCs). The theory cannot determine however if such scenarios are physically possible.

Awfully good point. Neither does special relativity. All you have to do is draw a closed curve and there you have it. So Dale and Rasalhague are talking about the continuous Lorentz transformation of a vector attached to a single point on a world line. (These are transformations within the Lorentz subgroup (SO+(3,1).) If the vector is postive-timelike to start with, it is always positive-timelike. Classically, over the world line of a particle, no matter how the energy of the particle is changed, from point to point, in a continuous manner, the displacement vector will remain positive-timelike.

I hadn't thought to consider what sort of discontinuous changes in energy or mass a particle would need to undergo to jump to the other sheet of the hyperboloid. These transformations are improper Lorentz transforms of positive determinant.

 

[Dale]

General relativity does not exclude the possibility of closed timelike curves (CTCs).

However, even in a closed timelike curve you can never have a future directed timelike worldline turn into a past directed timelike worldline.

 

[Dale]

If "other observers" is synonymous with "other coordinate systems", doesn't this make it a coordinate dependent measurement of distance like any other? Well, maybe it's just a matter of words, but if we can adopt a special definition of distance in this instance, why not in any instance, and just define a particular distance measured in a particular way as "invariant". But I suppose that would undermine the distinction between "coordinate dependent" and "invariant", causing us to have to find new words for those concepts, which would be a nuisence...

Oops I may have acquiesed to Passionflower's criticism too early.

In the sense that I mentioned the quantitative results of ALL physical experiments are frame invariant. The question is whether or not the experimental result is distance in a given frame. Although the numerical result of a radar-time experiment is agreed upon by all reference frames, equating that result to distance is only correct in the frame where the clock is at rest.

 

[Dale]

Awfully good point. Neither does special relativity. All you have to do is draw a closed curve and there you have it.

This is not correct. Such a curve is indeed closed, but it is not timelike. Closed timelike curves (CTC) are NOT possible in SR.

 

[Phrak]

This is not correct. Such a curve is indeed closed, but it is not timelike. Closed timelike curves (CTC) are NOT possible in SR.

Perhaps not, but I confused definitions in any case. Feel free to replace references to CTC in my past, above, with just 'loop'.

 

[Max™]

Don't think of time as a process, or a flow. Time is an axis against which you can measure differences in spatial components, be they extended objects, or spatial dimensions themselves.

 

[Yuripe]

Don't think of time as a process, or a flow. Time is an axis against which you can measure differences in spatial components, be they extended objects, or spatial dimensions themselves.

Time would be then one dimension of spacetime, where theoretically you can "move" forward of backward by dt as a vector.
Direction of this vector distinguishes if it's forward or backward and it's length describes the rate of this "movement".
This is as viewed from a perspective of the observer confined to this one time dimension.
In reality we see only dt>0 so it would seem time is unidirectional.

Another thing is a use of terms dependent of time to describe time, like moving, speed, etc. It's hard to talk about time or spacetime and use terms that are independent of them.
Sometimes we even do it by looking at time or spacetime from a perspective of extra added dimension(s).
If you look at spacetime, there is no movement in it, everything in it is placed and stationary.

Another interesting thing that comes to my mind as I read along is an idea of worldline for a tunneling particle.
Experiments with light show that while it's tunneling through space, the light travels at speeds at least a few times grater that normal speed of light.
I think the same goes in other types of tunneling for example when atom absorbs light, electron changes it's orbital and this change occurs by tunneling.
Do these particles cross wordline or just temporarily "detach" from spacetime?

 

[Max™]

Charge seems to effect your motion through time, and any motion through space, be it possessing mass, being near a gravity well, or accelerating, affects your motion through time.

If you look at spacetime, there is no movement in it, everything in it is placed and stationary.

Yup, your awareness is a process which updates itself to note the last time it updated itself, giving you the persistent illusion that now is the only moment that has ever existed.

You felt that way a second ago, you'll feel that way after you finish reading this sentence.

Those moments still exist, over there *points towards the past*, or are waiting for you to see them over there *waves a hand in the direction of the future*.

 

[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.