I Temporal length vs. Temporal displacement

[LeeE]

TL;DR Summary

Is temporal length distinct from temporal displacement?

While we regard spatial displacement and spatial length as distinct from each other we don't seem to treat time in the same way. Instead, when anything refers to temporal length it seems to actually be referring to temporal displacement.

For example, a 5 metre long automobile may travel 100 km but will still be just 5 metres long once it has arrived. This distinction just doesn't seem to appear in the temporal realm though, which doesn't seem right to me.

Furthermore, it seems to me that this lack of distinction may raise some important issues; if we regard the temporal length of the Universe as being its age then should we not be multiplying the total mass/energy of the Universe (as we currently think of it) by that temporal length to get its total space-time mass/energy? If so, then where is the new mass/energy coming from?

Conversely, if we regard the temporal length of the Universe as being distinct from its temporal displacement (its age) then that problem goes away; the total mass/energy of the Universe stays the same, with no need for new mass/energy to come from anywhere.

The idea of temporal length being distinct from temporal displacement also raises some interesting implications. For example, if the temporal length of the Universe is distinct from its temporal displacement, or age, then time travel would not seem to be possible, at least within our Universe, because our Universe would no longer be where the time-traveler went back to; it will have moved on. But, on the other hand, if our Universe has a distinct temporal length, then it is possible for other universes to exist ahead of and behind us, so time travel to a different universe might be possible. The same would apply to Worm-holes too.

The more I think about the idea of temporal length being distinct from temporal displacement, the more sense it seems to make to me, but I can't seem to find anything anywhere about the topic. Surely, other people must have looked into it so I'm a bit surprised that I can't find any discussions about it. There seem to be quite a few interesting implications but I can't find anything to follow-up on them.

Can anyone shed any light or thoughts on this?

 

[Ibix]

You are over-complicating something simple. Length and distance are the same thing just in different contexts. In your example of a car, the distance it travels is the length of the road.

In terms of time, the analogy is something like a 2 hour film shown daily at 3pm. The film is two hours long both days, and there are 22 hours between the two showings.

The rest of your speculation looks a lot like you need a good physics textbook, I'm afraid.

 

[Dale]

If you think about a human in spacetime we are a “linguini noodle” shape. Roughly 6 ft wide, 2 ft thick, and 2.3 10^18 ft long. We are oriented so that our length is along the time direction.

You can slice the noodle across its width at two different times and project those slices onto space. The distance between the projected slices is the spatial displacement you mention.

To talk about a temporal displacement you would slice it along its length across two different positions and project those slices onto the time axis. Since we are so much longer than our width, those slices will almost entirely overlap. The resulting temporal displacement would be rather arbitrary.

It isn’t so much that temporal length and temporal displacement are fundamentally different from spatial length and spatial displacement. It is more that the objects we are interested in are shaped very asymmetrically. The uselessness of temporal displacement is due to that asymmetric shape

 

[Herman Trivilino]

TL;DR Summary: Is temporal length distinct from temporal displacement?

For example, a 5 metre long automobile may travel 100 km but will still be just 5 metres long once it has arrived. This distinction just doesn't seem to appear in the temporal realm though, which doesn't seem right to me.

If we watch a movie that has a temporal length of 2 hours, and 7 weeks later (a temporal displacement) we watch it again, it will still have a temporal length of 2 hours.

I don't see a distinction.

 

[LeeE]

Thanks for the responses. I thought it would be apparent from the question that I was familiar with the 'linguini' model and I used to think in those terms until I started looking more deeply into relativistic time-dilation, where the sum of two rate of change of displacement vectors i.e. the spatial & temporal velocities, equals 'c'.

If you normalize 'c' = 1 then v_t = sqrt(1-v_s**2).

(Normalizing 'c' = 1 also has the benefit of removing the rather unbalanced human size scales on everything. For example, on a human size scale 1 metre and 1 second are convenient - but there's a factor of 3E8 between them. We humans tend to think of the speed of light as being very fast but at the size scales in which the Universe operates it's really rather slow)

But, for example, if you observe a particle traveling at 0.5 'c' then its v_t will be 0.866. Thus, for every 1 second that passes for the observer only 0.866 of a second will pass for the traveling particle (if you plot the function you'll get a nice quadrant of a circle, from which it becomes apparent, if you hadn't already realized it, that both the spatial and temporal velocities can be defined as an angle in space-time - the v_s of a particle traveling at 0.5 'c' corresponds to sin 30 and the v_t to cos 30 (both from the temporal axis).

So while space and time appear to us to be entirely different things, at least with regard to how we are able to move through them, they are evidently similar enough that their velocities can be summed to give a meaningful result.

Of course, having gone that far, one can't avoid also thinking about relativistic foreshortening (in the spatial realm) and wondering if there is a corresponding phenomenon in the temporal realm, which would thus require a finite temporal length. This then prompts the question: what size is it?

Btw, if v_t = sqrt(1-v_s**2) then, when considering gravitational time-dilation, v_s = sqrt(1-v_t**2) which, on the face of it, is problematic for several reasons; v_s is scalar and has no 'direction', so there doesn't seem to be a way of expressing it (there's an energy issue too). Actually, I think that there may be a solution to this but it's very speculative and depends upon the concept of a very small temporal length.

 

[Ibix]

If you normalize 'c' = 1 then vt = sqrt(1-vs**2).

Here you seem to be defining $v_t=d\tau/dt$ and $v_s=dx/dt$.

(if you plot the function you'll get a nice quadrant of a circle, from which it becomes apparent, if you hadn't already realized it, that both the spatial and temporal velocities can be defined as an angle in space-time - the vs of a particle traveling at 0.5 'c' corresponds to sin 30 and the vt to cos 30 (both from the temporal axis).

So, here you are thinking of plotting $\tau$ on one axis and $x$ on the other, with $t$ as a parameter. That's an Epstein diagram (a space-proper-time diagram), or a close relative thereof. They're not particularly useful because you can't represent spacelike vectors or null ones, and two vectors added together don't necessarily point to the same place as their sum.

Minkowski diagrams are more popular for good reason.

 

[LeeE]

Minkowski diagrams are good for plotting space-time displacements and showing the space-time 'history' of a particle.

But relativistic effects occur due only to the rate of change of those displacements i.e. the velocities, and not the displacements themselves, so no, I wasn't trying to produce a Minkowski diagram; the plot just shows the relativistic relationship between those two velocities in our Universe that applies to everything within it. Thus, everything could be said to be traveling at 'c' through space-time (at least within its own frame of reference - a nominally 'stationary' observer who tries to sum their v_t with the v_s of a moving particle will always arrive at a figure of > c, but then they're mixing frames of reference).

So yes, it's just the relationships and how a change in one realm has a consequence in a different realm that interests me; not where and when a particle might think it is, but how it thinks it's 'moving'.

 

[Ibix]

no, I wasn't trying to produce a Minkowski diagram

I didn't say you were. I was just pointing out the well known limitations of the approach you are taking. The slightly more high falutin' way of putting it is that by differentiating with respect to coordinate time you are wedding yourself more tightly to an arbitrary coordinate choice and moving away from relativistic invariants which are the directly measurable quantities in the theory. It's not wrong to do so, but it's adding layers of interpretation on top of an already complex topic - that you get multiple distinct representations of one thing on the same diagram, and that you can't represent spacelike vectors at all, are symptoms of that. It's also a much less general approach - it won't work at all in coordinate systems with no timelike coordinate, nor ones where different coordinates are timelike in different regions. And it will not generalise at all well to general relativity. You're pretty much confining yourself to studying only timelike vectors expressed only in orthonormal coordinates on flat spacetime, which seems to me to be rather an odd limitation to self-impose.

 

[PeterDonis]

relativistic effects occur due only to the rate of change of those displacements

There is no such thing as a rate of change of a spacetime displacement. Relativistic effects occur because of spacetime geometry.

 

[LeeE]

You're pretty much confining yourself to studying only timelike vectors expressed only in orthonormal coordinates on flat spacetime, which seems to me to be rather an odd limitation to self-impose.

This is true. I'm not so much trying to understand how our particular 4-d space-time works but how n-dimensional space-times might work, especially 1, 2 & 3 dimensional space-times. Higher dimensional spate-times aren't really that much more difficult to work with but are much harder to visualize.

A 1-d space-time (the single dimension being time) seems to be a bit of an exception because it doesn't have scope for relativity - there's only the temporal vector - but the single sizeless particle that would occupy it would be capable of change over time (no, I'm not interested in what that change might or could be - just whether change is possible), so it could be regarded as a 'working' universe.

As soon as you get to 2 & 3 dimensional space-times though, then there's scope for relativity to play its part, both with regard to movement but also with the admittedly odd concepts of 1 & 2 dimensional mass.

But I think that all this is getting away from the reason I originally posted here. And to be clear, although I mentioned time travel in that post it wasn't the mechanism of time travel that interested me but where you might find yourself if you did as an example of one of the possible consequences of a universe with a short temporal length.

I should make it clear that I don't believe that this is how the Universe works - nobody does (even those who think they do) - all I'm doing is following a path of thought and reasoning that seemed to appear in the landscape of theoretical physics. I'm hoping that I've not embarked upon a wild goose chase and ended up barking up the wrong tree, in the wrong forest, having forgotten the fact that geese don't live in trees, no matter how annoyed they might be.

 

[PeterDonis]

I'm not so much trying to understand how our particular 4-d space-time works but how n-dimensional space-times might work, especially 1, 2 & 3 dimensional space-times.

This sort of discussion is (a) out of scope for this forum (relativity deals with our actual 4-dimensional spacetime, not hypothetical ones of fewer dimensions) and (b) out of scope anywhere on PF without some kind of reference as a basis for discussion.

Thread closed.