The Principle of Equivalence 53AgCBFigure 3.3A pocket lamp at ‘A’ in the Einstein lift is shining horizontally on a point ‘B’. How-
ever, an outside observer who sees that the lift is falling freely with acceleration푔concludes
that the light ray follows the dashed curve to point ‘C’.
However, the observer outside the tower sees that the lift has accelerated while the
light front travels across the lift, and so with respect to the fixed frame of the tower
he notices that the light front follows a curved path, as shown in Figure 3.3. He also
sees that the lift is falling in the gravitational field of Earth, and so he would conclude
that light feels gravitation as if it had mass. He could also phrase it differently: light
follows a geodesic, and since this light path is curved it must imply that space-time is
curved in the presence of a gravitational field.
When the passenger shines monochromatic light of frequency휈vertically up, it
reaches the roof height푑in time푑∕푐. In the same time the outside observer records
that the lift has accelerated from, say,푣=0togd∕푐,where푔is the gravitational accel-
eration on Earth, so that the color of the light has experienced a gravitational redshift
by the fraction
훥휈
휈
≈푣
푐
=
gd
푐^2=GMd
푟^2 푐^2. (3.1)
Thus the photons have lost energy훥퐸by climbing the distance푑against Earth’s grav-
itational field,
훥퐸=ℎ훥휈=−
gdh휈
푐^2, (3.2)
whereℎis thePlanck constant. (Recall thatMax Planck(1858–1947) was the inventor
of the quantization of energy; this led to the discovery and development ofquantum
mechanics.)
If the pocket lamp had been shining electrons of mass푚, they would have lost
kinetic energy
훥퐸=−gmd (3.3)climbing up the distance푑. Combining Equations (3.2) and (3.3) we see that the pho-
tons appear to possess mass:
푚=