bernhard.rothenstein said:
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[/color]"), 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.
MeJennifer said:
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.)
