Gravity bends light. One can argue this by reasoning that a beam of light should travel in a curved path when viewed from within an accelerating frame of reference, and then invoking Einstein’s principle of equivalence, which asserts that the effects observed in an accelerating frame of reference are indistinguishable from the effects observed in a uniform gravitational field. If light bends in an accelerating reference frame, it must also bend in a gravitational field.
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8.
The distance between Alice and Bob does decrease very gradually as V approaches c within the initial starting frame, which implies a small difference in velocity (at some point in time) within this frame; which, in turn, seems to imply a non-zero difference in time dilation. However, because Alice and Bob both have an initial velocity of zero, and the same final velocity, if V = dV —a substitution we will later make—we can assume their clocks remain synchronized within the initial starting frame over the interval dV. Put differently, the velocity difference that produces the length contraction in the rest frame is too small to take the clocks out of synchronization in this frame.
9.
The infinitesimal “
proper
” change in velocity (dV) is equal to the infinitesimal change in velocity as measured by the rest frame (dV) when Vi _ Bob = 0.
10.
This result turns out to be consistent with other, mathematically more complicated, derivations. See, for instance, http://www.physik.uni-leipzig.de/∼schiller/ed.
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Arguments for gravitational time dilation are sometimes based on gravitational redshift, as opposed to accelerational Doppler shift. These arguments aren’t erroneous, per se, but they aren’t totally convincing—as they often rest upon an unproven assumption that gravitational redshift can only be explained by gravitational time dilation.
19.
The entirety of the thought experiment, conceptually, is contained in the following four questions, all of which, I’ve found, students thoroughly enjoy discussing in small groups. All questions assume a basic knowledge of special relativity.
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