# Space-time and the speed of light

1. Oct 31, 2004

hi,
In special theory of relativity time is considered as a fourth dimension of space-time, using speed of light.I am worried, why speed of light should be involved in a theory of space-time ? Space and time must exist in the absence of light also. So do not we need a theory of space-time in which no speed of light comes ?
by
jp

2. Oct 31, 2004

### Staff: Mentor

If nothing happened in space, we wouldn't need a theory to describe it...

3. Nov 1, 2004

### Garth

Hi Jayantiprasad,  - And welcome to these Forums! (I've just noticed it was your first post)

"Time exists to prevent everything happening all at once; space exists to prevent everything happening at all the same place." (G.B)

Indeed events require a space and a time, for we cannot measure the position of anything except at a particular time and we cannot measure the time anything happens except at a particular place. This insight led Einstein to conceive a model of a four dimensional space-time continuum.

But how are these different dimensions, length, width, height, and time, to be connected? How can you measure the space-time separation or interval between two events?

Think first of two dimensions, the surface of a sheet of paper, for example, of length x and width y.
The shortest distance s between two opposite corners is given by Pythagoras' theorem:
s2 = x2 + y2
If we now go to three dimensions, the shortest distance between two opposite corners of your room, for example, one on the floor and the other in the opposite corner on the ceiling, where the room is x long, y wide and z high, is given by:
s2 = x2 + y2 + z2.
So what happens if we go to four dimensions, and measure the space-time interval between two camera flashes, for example, one happening at one corner of your room and other happening at the opposite corner but a few seconds t later?

We might think the answer would be:
s2 = x2 + y2 + z2 + t2, but we would be wrong. There are two things wrong with it.

First of all Einstein had been working on a problem, how to make Maxwell's equations independent of the observer's frame of reference, which would mean the velocity of light is equal for all observers as discovered by Michelson and Morley. He realised that these problems would be resolved if he adopted an idea of his colleague and lecturer, Minkowski, that the t2 term should be subtracted, not added.

Secondly we are adding "dollars and euros", the dimensions of the terms in the equation are not right, we need a conversion factor, "a rate of exchange", to convert one into the other before we can add or subtract them. The conversion rate that turns time into distance is a velocity we call it c, so if c is measured in kilometres/second and t is so many seconds, then ct will be so many kilometres.

So the correct equation giving the space-time separation s between two events becomes
s2 = x2 + y2 + z2 - c2t2, and now there is one more refinement to make.

Rather than measuring the distance across an extended interval of space-time, it is important to deal only with the separation of adjacent events separated by a infinitesimal change in the coordinates, dx, dy, dz, dt. This allows the possibility that space-time might be 'curved', just as the surface of the Earth is curved into a sphere, although it looks flat on a small scale, or alternatively like the surface of a saddle. Or perhaps like that of a Popperdom, all 'hills and hollows'. So the correct expression of the infinitesimal separation of two adjacent events is:

ds2 = dx2 + dy2 + dz2 - c2dt2, this is called the metric of flat space-time. If we want to include 'curved' space-time then we put a coefficient, not equal to one, in front of each term dx2 etc. In order to find out the separation between two events such as a distant super nova exploding and the event being observed on Earth you have to add up, or integrate all the infinitesimal ds intervals along the light path.

So you can see that the answer to
is that a velocity has to be built into the geometry of space-time to adjust the dimensions of the terms in the equation for the metric. Now it so happens that when we work out the kinematics of moving particles using this metric we find that for a moving object the mass measured by another observer increases as the velocity relative to that observer increases. When the velocity approaches the velocity c the mass approaches infinity, in other words no particle with rest mass can travel at the velocity c. However, photons of light have no rest mass and in a vacuum they have to travel at this velocity. Therefore the velocity c is the velocity of light in a vacuum, and we find it has to be built into the geometry of the space-time continuum.

Garth

Last edited: Nov 1, 2004
4. Nov 1, 2004

### Chronos

Brilliant

A brilliant explanation, Garth. You should be writing physics textbooks.

Last edited: Nov 1, 2004
5. Nov 1, 2004

Dear Garth,
I am thankful for your explanation. Few days back I asked same question to
David Deutsch, let me put his views here

> Q. In the special theory of relativity, time is
> identified as a fourth dimension of space-time with
> involving light there. I do not know why space and
> time can not exist without light (electromagnetic
> radiation) ? why we need light in a theory of space
> and time ?

The constant relating the units of space and time is known as the
'speed of light', but logically it is not derived from any theory of
light, but vice versa. If it were found tomorrow that light actually
travels at a different speed, we might want to find a new name for the
constant used in the theory of spacetime.

Yours sincerely
-- David Deutsch

From this answer few things are clear
1. The cofficient needed to balance the dimensions of space and time need not be the speed of light. In principle it could be speed of anything.
2. Speed of light need not be constant, because it is not derived
from any fundamental principle.
http://arxiv.org/abs/astro-ph/0305457

bye
jp

6. Nov 1, 2004

### jcsd

Jay,

1) correct, up to a point. If we talk about spacetime in a genrealized manner there is no mathematical reason that this constant should correspond to the speed of light.

2) Howvere the results of experiment pretty much demand that it is c. I've never been exactly clear on this, but I think the first postulate of relativty is at the very least suggestive of the second.

7. Nov 1, 2004

### Garth

Dear jp,
Yes, in my explanation (sorry if I was teaching my grandmother to suck eggs! I have just read your third post) I too pointed out that the conversion factor c is just a velocity.
However massless 'particles' seem to have to travel at this velocity. Where mathematics yields to physics, is over the question, "Do photons behave as (rest) massless particles?" I believe they do.

Fundamental constants are all related by the way we measure them. i.e. one may vary but if we define that parameter to be constant then our experiment will indicate that another is varying, to compensate. I believe this is what is happening with the Variable Speed of Light VSL theories. It may be the fine structure constant that is actually varying, but then what do I mean by 'actually'? It is a question of definitions and axiomatic adoption of a hierarchy of which constant is more 'fundamental' than the others.
(A hierarchical fundamentalism?? sorry I couldn't help that one!)

I choose to define c to be constant, it is a measurement thing - define c constant and then make time the fundamental measurement. In this case if c is observed to vary the FSC varies. Others may choose the other way round.

Chronos - you are embarrassing me - thank you! I'll write the physics text book when I know what physics it is that I have to write about!
Garth

Last edited: Nov 1, 2004
8. Nov 1, 2004

### DaveC426913

1] The structure and measurement of space and time are closely linked to a cosmic constant, which we've c.

and then:

2] By the way, one other thing that is dependent on this cosmic constant is the speed at which photons (*all* forms of electromagnetic radiation) travel.

i.e. The constant 'c' is more fundamental to the universe than the 'mere' speed of light.

9. Nov 1, 2004

### Garth

You took the words right out of my mouth!

Garth

10. Nov 1, 2004

hi,
I am again reading Minkowski's 1908 paper(space and time), very soon I will put key points. Let us look at the few lines

"Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality"

"Nobody has ever noticed a place except at a time, or a time except at a place"

"To establish the connexion (between space and time), let us take a positive parameter c "

"As a matter of fact. natural phenomenon do not possess an invariance with the group G_infinity (c=infinity), but rather with a group G_c(c=finite)"

With developing some logic then he put the axiom

"The substance at any world-point may always, with the appropriate determination of space and time, be looked upon at rest"

Then se says

"The axiom signifies that at any world point the expression
c^2dt^2 - dx^2-dy^2-dz^2 always has positive value"

and it implies v is always less than or equal to c

I will be back
by
jp

Last edited: Nov 1, 2004
11. Nov 2, 2004