# Basic question about the speed of light

1. Sep 20, 2010

### elmarto

Hi everyone,
I recently read an essay by Isaac Asimov called "Figure of the Farthest", which discussed the ideas on size of the universe across history.

One of the basic ideas of that essay was that, since objects move away from us at increasing speeds the farther they are, then when you observe an object that is moving away from us at a speed close to that of light, then you would effectively be seeing the 'edge of the universe' (the observable universe, at least).

In this essay, Asimov claimed that there could be many other objects beyond that edge, but we would never be able to see them because they move away from us faster than the speed of light. He then said that the "nothing can travel faster than the speed of light" claim is just a "simplification". The correct statement is that, every time you measure the speed of some object, it will always be lower than the speed of light.

So my question is, is all of this correct? I haven't been able to find any discussion about this, and nobody claiming that it's a simplification to say that nothing can travel faster than the speed of light.

Is it possible that there are things that travel faster than the speed of light, but just can't be observed because their light (or radiations or whatever) never reach us?

Thank you very much.

2. Sep 20, 2010

### Drakkith

Staff Emeritus
As far as current science is concerned, nothing with mass can travel faster (or as fast) than the speed of light. Ever. Note, that you would measure the speed from a nonmoving frame. IE if 2 objects are traveling at .75c opposite from each other, each would appear to the other as traveling faster than light away from each one. But in reality, each is simply moving at .75c.

3. Sep 20, 2010

### granpa

the galaxies arent really moving faster than the speed of light. The space between us and them is just expanding. As a result light from those galaxies far enough away will never reach us.

4. Sep 20, 2010

### Staff: Mentor

Actually, their speed with respect to each other would be about 0.96c.
The speed of an object, of course, depends on the frame doing the measuring. With respect to some specific frame (the earth, say) the two objects are moving at 0.75c. But with respect to some other frame, they'd be seen as moving at some other speed, but always less than c.

5. Sep 20, 2010

### JesseM

In special relativity nothing can travel faster than light in any inertial frame, but objects can have a coordinate speed greater than c in an arbitrary non-inertial coordinate system. However, in the curved spacetime of general relativity, it's impossible for any coordinate system covering a large region of spacetime to be "inertial", so in the type of coordinate system commonly used in cosmology, galaxies can indeed have a speed faster than light. It is still possible to have locally inertial coordinate systems in arbitrarily small regions of curved spacetime where the effects of curvature become negligible (see the equivalence principle), so in that sense it's still true that an observer in the same local region as a beam of light will find that in a locally inertial system the light moves at c and no object with nonzero mass in the same region can travel that fast.

6. Sep 20, 2010

### Drakkith

Staff Emeritus
Really? How would the speeds with respect to each other be 0.96c?

7. Sep 20, 2010

### Passionflower

There are three identical ways to calculate this, one is to use the (simplified) velocity addition formula:

$$v_{1+2} = {v_1+v_2 \over 1+ v_1 v_2}$$

As an alternative one can use rapidities:

$$\eta = \tanh^{-1} [v]$$

Rapidities can be added simply by:

$$\eta_{1+2} = v_1 + v_2$$

Finally we can use proper velocity (or celerity) which is useful as well as it tracks the coordinate distance over proper time, which is very practical if we consider particle collisions.

Proper velocity is:

$$w = \gamma v = c \sinh [\eta]$$

And proper velocity adds as follows:

$$w_{1+2} = c \gamma_1 \gamma_2 (w_1 + w_2)$$

It is interesting to consider the momentum with respect to velocity and proper velocity:

$$p = m \gamma v = mw$$

So proper velocity is simply momentum over mass:

$$w = {p \over m}$$

Last edited: Sep 20, 2010
8. Sep 20, 2010

### Drakkith

Staff Emeritus
Perhaps i am misunderstanding what you mean when you say that the velocities "Relative to each other". From an observer at a standstill, the distance between the 2 would increase at a rate greater than c correct?

9. Sep 20, 2010

### JesseM

Of course "at a standstill" can only be defined in a relative way, but in the frame of an observer who measures each of them to be moving at 0.75c in opposite directions, the distance between them will increase at a rate of 1.5c. However, in the rest frame of either of those observers, the other one will be moving away from them at 0.96c.

10. Sep 20, 2010

### Drakkith

Staff Emeritus
So each one will see the other receding away at 0.96c then? Hrmm...interesting...

11. Sep 20, 2010

### DaveC426913

Locally.

Distant objects can be - and are - receding from us at > ftl.

12. Sep 20, 2010

### Drakkith

Staff Emeritus
But they arent moving faster that light, correct? Thats simply a consequence of space itself expanding.

13. Sep 21, 2010

### elmarto

So, from what I gather from JesseM's and DaveC426913's replies, objects that are very distant from us can indeed move faster than the speed of light with respect to us (be it because they 'move' or because space is expanding). If I understand correctly, those objects would not be observable by any means, though. Do I have this right?

This also leads me to a more 'philosophical' question that is sort of related to this issue. Is it possible that the speed of light is 'important' only because we use our eyes to observe things, and our eyes work with light?

For example, if we didn't have the sense of sight and we'd have to rely only on our ears, wouldn't we conclude that no objects can move toward us faster than the speed of sound, because such objects would be unobservable?

14. Sep 21, 2010

### DaveC426913

No.

It is not merely the speed limit of light; it is the entire electromagnetic spectrum (radio through cosmic rays) - all energy.

Further, nothing - not matter, not energy, no information of any kind - can move faster than c.

It is tied to the very shape of spacetime and is the fundamental speed limit of the universe itself.

15. Sep 21, 2010

### PhilDSP

I've always had a problem understanding why a person would go from:

To:

In doing that the assumption is that no other energy or information can possibly exist other than EM energy or at least that plays by the same behavioral rules. True, energy manifestations that appeared to be diverse in the 19th century turned out to be electromagnetic as Maxwell showed. But isn't it risky to make the simplifying assumption above?

16. Sep 21, 2010

### DaveC426913

What exactly is the 'risk' of going forward on the evidence from the large body of knowledge we currently have?

If evidence presents itself to show that it is too narrow a scope then we explore that, but to qualify every step forward with 'it might be the wrong step, maybe we shouldn't move forward' doesn't get us very far, does it?

In short: no form of matter we currently know of - nor any form of energy we currently know of, nor any form of information we currently know of - can move faster than c.

Is that better? It shouldn't be; it goes without saying. All statements in science are qualified with 'that we know of'.

17. Sep 21, 2010

### Passionflower

No, because the farther the light goes from those objects into our direction the less the relative expansion factor will become.

18. Sep 22, 2010

### PhilDSP

As I see it, the risk of being too dogmatic in making certain pronouncements is that it can lock the door to imagination. And imagination and inquisitiveness has at times been the principle driver of advances in Physics and Mathematics. Without imagination it's pretty hard to discover something that can't be seen or register on some current technology instrument.

19. Sep 22, 2010

### DaveC426913

First, learn the material. Then learn to use it imaginatively.

Imagine if, in grade 1, we were given the number line and, rather than learn that 2+2=4, we decided to jump directly to "using our imagination" to perform arithmetic.... "The sum of all the curvy numbers should equal the sum of all the liney numbers."

Would you say 2+2=4 is "too dogmatic" and would "close the door" for later studies in, say, calculus or geometry?

20. Sep 22, 2010

### jambaugh

Thought I'd add a point. Initially c represented the speed of light (and all e-m waves) in vacuum and still does to the best of our ability to measure. But it is now more a matter of a unit conversion constant allowing us to convert between e.g. seconds and meters now that we think of space and time as parts of a unified whole.

Imagine sailors measuring depth in fathoms and lateral distances in nautical miles, to use the same rope to measure either or at an angle they need to agree on a unit conversion say k=fathoms/nautical mile. Their Pythagorean theorem for oblique measurements will then have this conversion factor stuck in it:
length^2 = depth^2 + k^2 distance^2.
It simplifies greatly when we use common units.

So too with intervals between space-time events except the Pythogorean theorem becomes the indefinite Minkowski space-time metric.
d tau ^2 = dt^2 - dx^2/c^2 or ds^2 = dx^2 - c^2 dt^2.

Once common units are incorporated into SR then basically c=1. (one light-second per second). Speeds are then slopes of paths in a particular coordinate frame and unitless.

Note that we do not add slopes for stacked wedges except as an approximation for very very small slopes (small angle approximation tan(x)~x). We likewise do not add velocities in SR except for very very small velocities (the Galilean approximation tanh(x)~x). We rather rotate in Euclidean space or "hyper-rotate" (psuedo-rotate) in Minkowski space-time using hyperbolic trig and boost parameters as pseudo-angles.

In Euclidean space its the angles we add when stacking wedges:
m' = tan( alpha + beta) where m1 = tan(alpha) and m2 = tan(beta).

In Minkowski space-time its the boost parameters we add when composing velocity shifts (boosts):
v' = c tanh(a + b) where v1 = c tanh(a) and v2 = c tanh(b).

The typical velocity addition formulas then are obtained by applying hyper-trig identities.

Then finally there is the empirical question of how fast light travels and in SR this equates to whether electromagnetic fields carry an intrinsic mass. To the best of our measurement light is massless and travels at 1 light-second per second. So it's still OK to call c "the speed of light".