# Mathematically rigorous definition of time ?

1. Dec 23, 2004

### roger

Hi

I'm looking for the mathematically rigorous definition of time ?

can anyone help me ?

Thanks

Roger

2. Dec 23, 2004

### ahrkron

Staff Emeritus
Time is usually taken as given in the description of physical processes. It is basically "used" instead of "described".

However, if you really want to look at the behavior it has in non trivial situations, and when its flow can be affected, the way to go is learning General Relativity.

3. Dec 23, 2004

### roger

I have also read that time slows, mass increases, length decreases when going faster than light....

but why do these things happen ?

Could someone explain in a fairly simple way ?

thanx

Roger

4. Dec 24, 2004

### ahrkron

Staff Emeritus
In a way, there is no "why". The universe behaves in some way, and all we can do is describe it.

That said, it is sometimes possible to find unifying principles that help understanding some phenomena.

In the case of the effects you mention, they basically come from the fact that what we call "time" is affected by the fact that there is a maximum speed in nature. This speed is high enough to have allowed us develop a world view that assumes it to be infinite.

However, when you take it into account carefully (which is what Einstein did), you find out that "time" and "space" are not quite what your experiences seemed to imply.

Unfortunately, I don't have the time to summarize all relativity here, but I'd advise you to get a good book on it (you may find a good one on the physics napster in the forums), and come back with questions.

5. Dec 24, 2004

### xck

does anybody really know what time it is?

"mathematically rigorous definition of time ?"

the best definiton of time i have experienced is when my face came off my watch >wink< it was rigor, rigormortis...sorta

6. Dec 24, 2004

### roger

when you mentioned above that the speed of light is the maximum, what exactly is speed ?
What I'm trying to ask is what is motion and /or speed ?

Why can't we go faster than the speed of light ?
I read that your mass increase but how ?

Roger

7. Dec 24, 2004

### Rob Woodside

Motion happens when something changes place. Newton said "time is defined to make motion look simple." A little calculus cleans up the idea of distance travelled divided by time taken to that of speed or with direction, velocity.

Apart from it being an experimental fact, there are lots of reasons. The causal principle that causes should be observed before their effects would be violated with superluminal speeds. Rest mass, m, is the mass measured in the body's rest frame. The energy of a body is E = mc*c + K, where K is the Kinetic energy. In Newton's mechanics K = (1/2)mv*v, but in special relativity K becomes infinite as the speed approaches c. Thinking that E = Mc*c so that M is some kind of relativistic or apparent mass, you get M going infinite as the speed of light is approached. Notice that rest mass, defined in the rest frame, is still the same rest mass in any frame.

Read some relativity, you might enjoy it.

8. Dec 24, 2004

### pervect

Staff Emeritus
The best defintion I've seen of time is this:

time is what you measure with a clock.

This is an empirical definition, not a mathematical one. You can look at the SI definition of the second to find out what makes a "good" clock, and find that atoms make good clocks. Cesium atoms are mainly used because they are one of the most conveneniet "heavy" atoms above hydrogen in the periodic table, but basically all atoms make very good clocks. They make good clocks because they are very accurate, very precise, and very repeatable - and because, since atoms of a particular sort are all the same, there are no "manufacturing" issues.

Note that the sort of time you can measure with a clock is an interval - you define a path the clock (atom) takes, look at the reading of the clock at one point, and look at the reading again at another point. There is another definition of time that's a abit more sophisticated, this is called "coordinate time". To be able to define coordinate time, one must be able to synchronize distant clocks. To really be able to measure velocities, it turns out that you need coordinate time, which means that you need to be able to synchronize clocks. This is not as trivial as it might first appear, because it turns out there isn't any one single way to synchronize clocks! This rather suprising fact is one of the cornerstones of relativity.

9. Dec 24, 2004

### jdavel

roger,

You go first.

What's a mathematically rigorous definition of location?

10. Dec 24, 2004

### roger

a point in euclidean space

roger

11. Dec 24, 2004

### jdavel

roger,

Not bad! But "euclidean space" is an abstract concept. It's not the same as the physical space of the universe. Euclid could just as well have defined Euclidean time and put points in that representing particular times.

In the end, I don't think there's anyway to define space or time completely mathematically.

12. Dec 24, 2004

### roger

But why, these two things are the most common in physics and they appear in equations everywhere ?

What can be said about time and space in the quantum physics ?

Roger

13. Dec 24, 2004

### pervect

Staff Emeritus
Okay, then a mathematically rigorous defintion of space-time is a point in an Lorentzian space. If that helps......

14. Dec 24, 2004

### Rob Woodside

There is a difference between mathematical and physical theories. Mathematical theories have undefined primitives, like point and line, which acquire their meanings through their properties expressed by the axioms. A physical theory will ascribe a real correlate to the undefined primitives such as "the end of a plumb bob" for a point and a "light beam" for a line. Then one has a physical theory that stands or falls on experiment.

15. Dec 26, 2004

### roger

Can anybody else explain why mass increases and why you cant go faster than light ?

Does time exist for scalar quantities eg if there is no change in motion ?

16. Dec 26, 2004

### pervect

Staff Emeritus
Not being able to go faster than the speed of light doesn't really have anything to do with whether or not mass increases with velocity. In fact, the most commonly used form of mass, invariant mass, does not increase with velocity at all, but this is a separate question.

What is true is that if you use the relativistic velocity addition formula

vsum = v1+v2 / (1+v1*v2/c^2)

repeatedly, and add together any string of velocities less than 'c', your result will always be lower than c, no matter how many velocities you add together, as long as they are all less than c. No reference to mass is needed to derive this results.

What this means is that you cannot go faster than light by any form of accelerating in a normal flat space-time coordinate system where SR applies - note that this argument doesn't apply to cosmology, for instance the "Hubble expansion".

You can hypothesize hypothetical particles that never go slower than light (tachyons), which have very odd properties, and have never been observed.

There's one remaining pair of possibilities - wormholes & time machines. The former involve highly non-flat space-time, don't really involve motion through real space, and are subject to becoming time machines if one is free to move around their terminii (unless quantum vacuum fluctuations blow them up first, but that's yet another tangent).

17. Dec 27, 2004

### s3nn0c

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