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Tensors and Relativity: Chapter 5

# Non- existence of an inertial frame at rest on earth

## Non- existence of an inertial frame at rest on earth

If Special Relativity  is to be valid in a gravitational field, it is a natural first guess to assume that the ``laboratory'' frame at rest on earth is an inertial frame. Let us draw a spacetime diagram representing the above experiment [ see Figure 5.2 ]. We consider light as a wave, and look at two successive crests of the wave as they move upward in the gravitational field.

Figure 5.2: Minkowski geometry  for the Pound- Snider Experiment.

The top and bottom of the tower have vertical world lines in this diagram since they are at rest. Light is shown moving on a curved line, to allow for the possibility that gravity may act on on light in an unknown way, deflecting it from a null path. But no matter how light is affected by gravity the effect must be the same on both wave crests, since the gravitational field is not time dependent. Therefore the two crests ' paths are congruent, and we conclude from this hypothetical Minkowski geometry that

But we know that , and since the Pound- Snider experiment tells us that we know that . Therefore we have to conclude that our answer using Minkowski geometry is wrong!

• So the reference frame at rest on earth is not inertial!!

Is this the end of Special Relativity !!?..... Not quite. We have shown that a particular frame is not inertial, not that there are no inertial frames . We will find that there are certain frames which are inertial in a restricted sense, but before we define these frames let us first consider what we mean by the mass of a body.