By Firk F.W.K.

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**Additional resources for Age of Einstein (intro to relativity)**

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46 An observer at the position of clock 2 will conclude that the pulse of light coming from clock 1 had been emitted by a source moving downward with velocity v. The light is therefore “Doppler-shifted”, the frequency is given by the standard expression for the Doppler shift at low speeds (v << c): f′ ≈ f0[1 − (v/c)] = f0[1 − (al/c2]. The frequency f′ is therefore less than the frequency f 0. The light from clock 1 (below) is “red-shifted”. Conversely, light from the upper clock traveling down to the lower clock is measured to have a higher frequency than the local clock 1; it is “blue-shifted”.

If, under certain circumstances, photons behave like particles, we are led to ask: are photons affected by gravity? We have E PH = mPHI c2 = hf PH, or mPHI = EPH /c2 = hf PH /c2. By the Principle of Equivalence, inertial mass is equivalent to gravitational mass, therefore Einstein proposed that a beam of light (photons) should be deflected in a gravitational field, just as if it were a beam of particles. (It is worth noting that Newton considered light to consist of particles; he did not discuss the properties of his particles.

A1. Some useful mathematics: transformations and matrices Let a point P[x, y] in a Cartesian frame be rotated about the origin through an angle of 90°; let the new position be labeled P′[x′, y′] +y P′[x′, y′] P[x, y] -x +x -y We see that the new coordinates are related to the old coordinates as follows: x′ (new) = −y (old) and y′ (new) = +x (old) where we have written the x’s and y’s in different columns for reasons that will become clear, later. 1] = [5, 7] and so on. Some of these changes are shown below y y′ New axes and grid-lines are oblique x′ x This is a particular example of the more general transformation x′ = ax + by and y′ = cx + dy where a, b, c, and d are real numbers.