# From special relativity to the classical one

## Main Question or Discussion Point

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?

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HallsofIvy
Homework Helper
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.

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?
I can agree with one but not with two.

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

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

What does 'cover' mean? Or is this philosopy?
Primum vivere, deinde philosophari !
Cover in my poor English: Lorentz transformations become the classical ones!

I can agree with one but not with two.

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

HallsofIvy
Homework Helper
What does 'cover' mean? Or is this philosopy?
I interpreted it to me they are the same theory.
I can agree with one but not with two.

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

Dale
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.

DrGreg
Gold Member
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?
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.

I can agree with one but not with two.

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

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..

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

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...