The title pretty much gets the whole question across. In particular, did Einstein work with the modern definition of smooth manifolds, or was that formalized later? Did he know about/use the coordinate-free approach, or did he only consider vectors/tensors to be "things that transform as <rule>"?...

In classical mechanics we often define the action as the quantity $$ \int_{0}^{T} \left[ T - V \right] dt$$ Which in many applications is some variant of $$ \int_{0}^{T} \left[ \frac{1}{2}m \left( x' \right)^2 - V(x) \right] dt. $$ The usual justification for the principle of least action is the...

There is plenty of evidence for a cosmological time dilation effect. For instance a supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift $z=1$ (see Ned Wright's cosmology FAQ, Ned Wright on Tired Light). I understand how time dilation comes about wh...

The above problem is taken from the 2014 $F=ma$ contest. This question generated a great deal of controversy. At least two test takers challenged the answer, one who even tried to do the experiment. I’m told that several different “Ph.D” physicists declared that the correct answer was X, but, i...