# Time as a dimension

## Main Question or Discussion Point

This may be a dumb question, but if the universe is expanding and time can be treated as just another dimension, does this mean that time is 'expanding'? And if so, what the heck does that mean?

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pervect
Staff Emeritus
In standard cosmological coordiantes, time is not expanding. You could use coordinates in which one interval of coordinate time corresponded to different amounts of clock time, but there is no reason to do so.

There is a good reason to use coordinates in which an object from which the universe appears to be stationary with respect to the CMB has constant spatial coordinates, however. These sorts of coordinates have the property that the distance between objects with static coordinates expands as a function of time.

In standard cosmological coordiantes, time is not expanding. You could use coordinates in which one interval of coordinate time corresponded to different amounts of clock time, but there is no reason to do so.
Fair enough, I'm still puzzling over the time dilation thing, and I thought this might be related.

There is a good reason to use coordinates in which an object from which the universe appears to be stationary with respect to the CMB has constant spatial coordinates, however. These sorts of coordinates have the property that the distance between objects with static coordinates expands as a function of time.
Hmm. I thought maybe the... rate of expansion with regards to space might be related to an 'expansion' of time. But now I guess I'm muddling in the whole time slowing down as you get closer to the speed of light thing. Damn its easy to get bogged down in analogies.

Thanks for the help.

Joe,
Not a dumb question, but let’s split it into 3 parts.

Part 1 - ignoring expansion of universe, time dilation and quantum mechanics:

Is time a dimension in the sense that we can travel through it, just as we can travel through space?
1. Yes (but see 6. below).
2. We are all time travellers. Because you, your chair, computer, room, planet etc. are travelling at the same rate (at 1 year per year *) you don’t notice it.
3. The scary thing is that although the EVIDENCE of what you did a year ago may still be accessible (though in a ‘fossilised’ form), you cant go back to the REALITY of there because it was used to create what you are now. You may replace the word ‘year’ with atto-femto-second or Plank-time if you like. So in a sense, the dimension of time is being created continually.
4. When you think of a classical object moving in 1d space you can still think of it as being described by a Gaussian(x - vt)/a probability density whose centre moves classically - where a is the x-spatial extent of the object. (This is an intermediate step to thinking quantum mechanically.) You easily make this more general by multiplying by corresponding y & z Gaussians.
5. Likewise when you generalise to 4d you need to replace (vt) with (u \tau) and multiply by a Gaussian(t - u^0\tau)/b where \tau is proper-time at the centre of the classical object, b is the temporal extent of the object, u is the ‘spatial component of 4d proper-velocity’ and u^0 is the ‘temporal component of the 4d proper-velocity’ *. Don’t worry! you can skip 4 & 5 above.
6. Just as you have a spatial extent (a) you also have a temporal extent (b). The main difference between them is that in 3d the spatial extent can be revisited, but not the temporal one (as explained in 2,3 &5). But in 4d space & time are considered as one entity, so the 4d-spatio-temporal extent can not be revisited either.

* When you just sit there (rather than running around) u^0 = 1year/year = 1sec/sec.
I am afraid I have dropped into the convention of writing c = 1.

Part 2 - concentrating on spatial expansion of universe

1. Just as time is being created at 1 billion-years per billion years, so is space at about 0.6 billion-light-years per billion years. I pulled 0.6 out of a hat to make the following point. If you think the universe is hurtling through space now, then it is hurtling even more through time.
2. The 2d analogy (or parable) is that of an expanding balloon with small paper discs stuck on the surface. The discs are the galaxies (held together by their own gravity) that do not expand, while the space between them expands. So the recession of further galaxies is faster than that of nearer galaxies (Hubble effect or cosmological red-shift).
3. The time (of 1.) & space (of 2) are not expanding into anything (like the hot air & debris of an explosion into already existing space). This time & space are being created.

Part 3 - time dilation due to observing receding objects.

1. This is the stuff of standard Special Relativity course. There is an analogy with the Doppler effect with sound in air - but be careful as there is no corresponding medium for light (other than space-time itself). You may find the following helpful http://www.csupomona.edu/~ajm/materials/twinparadox.html [Broken]
2. The above red-shift is nothing to do with the cosmological red-shift of part 2 which is a GR effect.
3. Neither is it anything to do with gravitational red-shift due to light ‘climbing out’ of the sun or massive body - which is yet another GR effect.

Cautionary note: I have had these ideas so long now that they have become ‘common sense’ to me - and you know that Einstein did not think too highly of that.

Good luck with your quest, Mike.
$$m = m = m = m = m = m = m = m = m = m = m = m = m = m = m = m = m$$

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Joe,
Not a dumb question, but let’s split it into 3 parts.

Part 1 - ignoring expansion of universe, time dilation and quantum mechanics:
Agreed.

Part 2 - concentrating on spatial expansion of universe
1 and 3 are agreed.

2. The 2d analogy (or parable) is that of an expanding balloon with small paper discs stuck on the surface. The discs are the galaxies (held together by their own gravity) that do not expand, while the space between them expands. So the recession of further galaxies is faster than that of nearer galaxies (Hubble effect or cosmological red-shift).
Yes, but this example is not really 2d, if you're talking about a balloon. Maybe I'm abusing the example, but if the 2d space is expanding and the galaxies are moving apart, isn't the air filled part of the balloon equivalent to the past points in 'time'. The galaxies move apart as time progresses, as the balloon expands, but if the past still exists.... I suppose maybe in your example the present is all that exists... hmmm... that 3rd 'balloon dimension' is I guess what I'm thinking of.

Part 3 - time dilation due to observing receding objects.

1. This is the stuff of standard Special Relativity course. There is an analogy with the Doppler effect with sound in air - but be careful as there is no corresponding medium for light (other than space-time itself). You may find the following helpful http://www.csupomona.edu/~ajm/materials/twinparadox.html [Broken]
2. The above red-shift is nothing to do with the cosmological red-shift of part 2 which is a GR effect.
3. Neither is it anything to do with gravitational red-shift due to light ‘climbing out’ of the sun or massive body - which is yet another GR effect.
Yeah, this part I need to read more on for sure.

Cautionary note: I have had these ideas so long now that they have become ‘common sense’ to me - and you know that Einstein did not think too highly of that.

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Problems with the balloon thickness & radius.

Jo - I like the snap-shot of your brain ticking over - I hope the following helps.

1. You are right about the 2d-balloon 2-sphere 2-surface - it is just an analogy for 3d space (or 3-sphere) volume (3-surface). Only the surface counts in this analogy, you are ‘constrained’ to the surface.
2. If you think of the thickness of the balloon, as the ‘thickness in time’ or temporal extent (of Part 1.6) then it is a very easy but mistaken step to then think of the balloon radius as past time itself. (In fact it is such an easy step to make that you should probably forget about the balloon thickness all together.) Even if the balloon radius were past time, you would not be able to revisit it - see Part 1.3 and 7. If anything, the balloon radius is related to the Hubble ‘radius’ (rather than time - see 4 & 6). I made this mistake, but we are in good company - see 5.

3. Einstein thought of the universe as a closed 3-sphere (see 6). If you go on a straight line (geodesic) on (constrained to) the earth’s surface, you can’t fall off and you will always see the same curvature of the earth, there is no edge (boundary) to fall off. Similarly for the 3-sphere which explains why we don’t see more galaxies in one direction than in another (when you look at a billion-light-year scale).
4. But on this above model (see 6) the universe reaches a maximum radius (in say another 10 billion years), then contracts down to a big crunch 24 billion years later. Time would not start to run backwards after this 10 million years, so the balloon-radius is not time.
5. For a time Stephen Hawkin thought that the relationship between time & entropy would become horribly reversed when the universe started to collapse. He changed his views later. I don’t know enough about this, but I suspect he was making a more sophisticated version of our mistake.

6. Since the 1990’s there has been evidence that the universe expansion is not slowing down, but accelerating. This implies an ‘open’ universe. I gather the problem of the no-boundary is explained by it being beyond the ‘visible’ universe (i.e. light can’t reach us from beyond this ‘boundary’ anyway) - again I am out of my depth. It turns out that even in this open (parabolic) universe - forget the balloon, just consider 3 ordinary dimensions expanding - the further away galaxies are, the faster they separate. The Hubble ‘radius’ is then a bit more abstract.

7. When I started to learn SR, I got the impression that past (& future) time were still ‘sitting out there’ ready to be accessed by ‘fast moving’ objects. I suspect even Roger Penrose thought along these lines when he spoke on a TV or Radio program about the temporal extent of a so-called time-independent wave function in QM. (My gut feeling is that resulted from extrapolating a maths approximation and ignoring a Gaussian envelope of Part 1.5.) If you think about the reality of space & time of the ‘moving’ & ‘stationary’ objects rather than the graph of their world-lines (which can ‘sit out there’ in space-time), then the ‘real’ bit that counts is their ‘now’ bit (which is a more verbose version of your last sentence).

Good luck, Mike.
$$m = m = m = m = m = m = m = m = m = m = m = m = m = m = m = m = m$$

6. Since the 1990’s there has been evidence that the universe expansion is not slowing down, but accelerating. This implies an ‘open’ universe. I gather the problem of the no-boundary is explained by it being beyond the ‘visible’ universe (i.e. light can’t reach us from beyond this ‘boundary’ anyway) - again I am out of my depth. It turns out that even in this open (parabolic) universe - forget the balloon, just consider 3 ordinary dimensions expanding - the further away galaxies are, the faster they separate. The Hubble ‘radius’ is then a bit more abstract.
This makes me think of it more as an expanding ripple of water, that is flattening at the same time, ultimately causing a loss of surface tension and therefore accelerated expansion.

Gonna have to think more on this and probably read more too, but thanks a lot, lots of brain food here.