# Space, c and time question.

1. Nov 1, 2006

### gruff

Hi

In trying to get to grips with SR/GR (I'm still unsure about what differentiates them!) one of the things that springs to mind is what it means for the nature of time.

I've always thought (not too hard, just as a natural consequence of everyday experiance) that time is some fundamental property of the universe.

But, recently I've come to think of it as this: Since c is an upper limit for speed (not the right term I think) and space is assumed to be a fundamental quality of the universe, time becomes a direct consequence of these two things rather than a thing in itself.

Ie time 'passes' because it is not possible for a particle to change its position instantaneousely.

Is this a wrong way to think about time or just stating the obvious?

2. Nov 1, 2006

### robphy

[spacetime] curvature is the key difference between SR and GR.

Everyday experience could lead to that conclusion.
However, SR/GR (which is not yet part of everyday experience, but in agreement with precision experiments) tells us that spacetime (and not time) is a more fundamental property.

It's okay to think of c as "an upper limit for speed [of signal propagation]".
In a similar way as above, spacetime (and not space) is a more fundamental property. In SR/GR, time is a consequence of a specific type of curve for observers in spacetime---it is akin to the arc-length of that type of curve. Locally, space for an observer is a consequence of being perpendicular to [the tangent vector to] those curves. So, in SR/GR, space and time are consequences of structures in spacetime.

It is more correct to think of "[wristwatch] time passing for that particle" rather than "time" (which suggests the universal "absolute time" that exists in Galilean and Newtonian physics, but not SR/GR).

3. Nov 1, 2006

### gruff

I don't yet have the maths under my belt to understand completely what you mean by some of the terms in the above.

Can I get this point clarified though, it might help me: are the four dimensions of spacetime similar to eachother mathematically?

What I mean is, is it OK to think of them as similar to each other in the same sense that the three axes of cartesian space are similar to eachother?

Ah, hang on I think I get it.... so a point in space (eg an idealized point-like particle) is really a line (curve) in space time? Ie something extended in 2 dimentions in spacetime, not a point 'moving in time'.

And our personal experiance of time passing is akin to flipping through 3D slices of 4D spacetime?

4. Nov 1, 2006

### RandallB

Do not accept the idea that the 4 dimensions of GR can be viewed as a Space-Time solution dealing with 3 + 1 dimensions of x, y, z plus t. That is more inline with The Standard Model, QM, Strings, M, etc; where you might have 9 + 1D, 10 + 1D or some other variations to consider.
What Einstein in GR put forward is four dimensions better thought of as A, B, C, D all equivalent to each other. Thus something in that 4D world can “warp” just as we see things warp in x,y,z. It is that warp in the 4D; A, B, C, D dimensions that gives rise to both time and gravity as we see them in our 3D; x,y,z experience. That is the essential piece of GR that is so difficult for many (particle physicists) to accept it as completely correct. But Astrophysicists love GR because it gives such accurate answers.

You should find SR much easier to comprehend than GR, although the math for GR was created before Einstein it took him 8 to 10 years to work it out. So don’t expect a simple explanation (like 'think of time as just another dimension') to make GR clear, the math needed to understand GR is pretty challenging.

An additional note: there is a fundamental difference in thinking to use GR 4D (Also 5D GR options considered in the 1920’s) verses the particle theories of QM and others with or without multiple dimensions.
This is why most agree that GR and QM are fundamentally incompatible.

Last edited: Nov 1, 2006
5. Nov 1, 2006

### gruff

That's cleared up my one question thanks (I also looked on wikipedia in the mean time and saw some of the maths that shows SR treating the 4 coordinates the same (0,0,0,0).

6. Nov 1, 2006

### JesseM

They are not treated the same, though. In space, if you lay out some coordinate axes x, y, and z, and find the coordinates of two points in space, the distance between them is given by the pythagorean formula $$\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$$, where $$\Delta x$$ represents the difference in the two points' x-coordinate, and same for the others. The distance between points is coordinate-invariant, in the sense that even if you move or rotate your coordinate axes so that the coordinates assigned to each point is changed, when you plug the new $$\Delta x$$ and $$\Delta y$$ and $$\Delta z$$ into the same formula, you will get the same value for the distance between the two points.

In spacetime there is also the notion of an invariant "distance" between events (ie points in space and time) that is the same in different coordinate systems. But it is no longer found by just adding the squares of the differences in each coordinate--the sign of the difference in time coordinate must be the opposite of the sign on the position coordinates, so that the invariant "distance" between two events can be given by the formula $$\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - \Delta t^2}$$ (this is assuming you're using units where the speed of light is 1, like if you measure distance in light-years and time in years; if not, then it must be $$c^2 \Delta t^2$$ rather than just $$\Delta t^2$$). So you can't treat time as being just like another spatial dimension, if it was you'd use the four-dimensional pythagorean formula where you just add the squares of the four coordinate differences and the distance is the square root of that.

7. Nov 1, 2006

### pess5

Relativity shows that a time has no meaning w/o specifying a location in space, and vice versa. Also, the light ray travels some distance x per duration t in the one frame, and some distance X in some duration T in another frame of relative motion. So x/t=c=X/T where x<>X and t<>T. Clearly, motion cause a change in the measure of length, and since c is invariant, then with a change in length (X) must always come with an identical change in time (T). Hence, they are FUSED. It's a fused continuum of spacetime, coined by Einstein's math teacher Hermann Minkowski around 1909 or so.

Granted, we percieve space & time differently. Yet, they are fused into a single dimensional continuum.

pess

8. Nov 2, 2006

### gruff

Right so we just perceive them as different aspects of the same underlying thing (ie they 'resolve out' differently in experiance).

What about this anaolgy: 2D flatland vs the 3D world where the z axis (or y if you take y to be up) would seem very different to a flatlander to the x and y 'spatial' dimentions that they experiance. There is infact an x y z continuum (ignoring time for moment) - ie a flatlander would have the same conceptual difficulty understanding 3D space as we do understanding spacetime? I'm not confusing the two, just using them as an analogy.

Actually the analogy works even better if we imagine collapsing time into the z spatial dimention:

Time in flatland can be expressed as the entire xy flatland plane moving along the z axis (the flatlanders have no say in this!). At each 'step' (the steps are infinite though) objects in flatland are allowed to change their position in x or y only. Because they can't physically move in z they can't travel bacwards or forwards in z (ie time) so can't interfere with things or have bizare experiances like seeing their world from above. Their 'time' is essentially changes in xy with respect to z.

If this analogy is extrapolated to 3Dland, our 'resolving' spacetime(xyzt) as 3 of 'space' + 1 of 'time' can sort of be understood. Objects in 3Dland can freely chang position in their spacetime (xyz) with respect to the fourth spacetime dimension t in this same way. Makes sense?

So spacetime x, spacetime y, spacetime z, and spacetime t are all equivilent (same units etc), but our experiance of them is different because we are limited to only moving freely in spacetime(xyz) so it only appears that there are 3 'space' dimentions and a different one called 'time'?

Phew... bit wordy but that makes sense to me.

9. Nov 2, 2006

### gruff

Right. You can't treat the time dimetion like a space dimention, because space is not the same as spacetime and time is not the same as spacetime. But you can treat spacetime t the same as spacetime x y and z, you just use a different equation. Is this what you're saying?

10. Nov 2, 2006

### JesseM

I don't understand what you mean by "spacetime t" or "spacetime x y and z". Spacetime refers to the combination of spatial dimensions and a time dimension, you can't call the individual dimensions "spacetimes".

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