# From special relativity to the classical one

1. Oct 22, 2008

### bernhard.rothenstein

I find in the literature the following statements:
1. special relativity and classical one cover each other at slow speeds.
2. they cover when c goes to infinity.
Please tell me with which of them do you agree?

2. Oct 22, 2008

### HallsofIvy

Staff Emeritus
I don't see any reason why they are not both true. They are essentially saying the same thing. If "c goes to infinity", the ALL speeds are "slow" in comparison.

3. Oct 22, 2008

### MeJennifer

I can agree with one but not with two.

If C is infinite the causal structure of spacetime is different.

4. Oct 22, 2008

### Phrak

What does 'cover' mean? Or is this philosopy?

5. Oct 22, 2008

### bernhard.rothenstein

Primum vivere, deinde philosophari !
Cover in my poor English: Lorentz transformations become the classical ones!

6. Oct 22, 2008

### bernhard.rothenstein

7. Oct 22, 2008

### HallsofIvy

Staff Emeritus
I interpreted it to me they are the same theory.
I'm not sure what you mean by "causal structure of spacetime". It is true that all formulas of special relativity coincide with the classical formulas in the limit as c goes to infinity.

8. Oct 22, 2008

### Staff: Mentor

The classical limit is v<<c. Which is true for any finite v in the limit as c->infinity. So I agree with HallsofIvy, both 1 and 2 are correct.

When you take the limit c->infinity you recover the causal structure of Newtonian physics, with a past and future that is agreed-upon by all observers in any state of relative motion.

9. Oct 22, 2008

### DrGreg

I agree with both.

(2) is mathematically more accurate but (1) follows from (2) anyway.

I suppose the counterargument is that (2) is physically impossible in our universe, whereas (1) makes physical sense and compatible with experiment. But (2) is mathematically correct and is technically more precise.

For example, kinetic energy is given by

$$\frac{mc^2}{\sqrt{1 - v^2 / c^2}} - mc^2$$
$$= mc^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2} + O\left(\frac{v^4}{c^4}\right) \right) - mc^2$$
$$= \frac{1}{2}mv^2 + v^2 O\left(\frac{v^2}{c^2}\right)$$​

(using "big O notation"), as $c \to \infty$, or, equivalently, as $v/c \to 0$.

We get the Newtonian formula as $c \to \infty$, or less precisely by "ignoring $v^2/c^2$", or, inaccurately, "by ignoring $v^4$".

Note: it's better to say "Newtonian" (or "Galilean") rather than "classical" when referring to non-relativistic theory. "Classical" usually means "non-quantum", so relativity theory can be classical in that sense.

The causal structure of Galilean spacetime is different to the causal structure of Minkowski spacetime. No-one is claiming they are the same, just that the first is the limit of the second as $c \to \infty$. (That's my interpretation of Bernhard's question, allowing for the fact that English is not his native language.)

10. Oct 22, 2008

### Naty1

I take "cover" to mean "converge" or "become closer and closer"...and so it seems both statements are ok.

If lightspeed WERE to reach infinity, then interactions would be instantaneous...causality WOULD change a big heap!! Nothing would be outside our timelike horizon....we could "see" the entire universe...I think the heavens would be infinitely bright...in fact if light were instantaneous would there not be infinite energy reaching us from the universe and everything would be instantaneously obliterated...everything would disintegrate to energy..

11. Oct 22, 2008

### matheinste

Hello Naty1

This may be a naive view but i think no-one seemed to have a problem with causality before the speed of light was found to be finite. If one thing causes another then how quickly the cause causes the effect does not alter their order. As for infinite brightness, you would still only see the light from distant objects as it was emitted but without the time delay due to the finite speed of light. If one emission hapenned at one point at one time and at a later time another emission happened we would still see them at sepaerate times.

Matheinste

12. Oct 23, 2008

### JANm

Hello matheinste
Could it be that Naty1 refers to the paradox of Olbers: why are the nights dark? Astronomers use this law to prove lumpyness of matterdistribution. If stars would be equally distributed through space nights wouldn't be dark...

13. Oct 23, 2008