## Distinguishing between Acceleration and Gravity

The Equivalence principle tells us that gravity and acceleration are indistinguishable. This principle follows from the consideration fo the experiences of a person sealed in a opaque box.

In 1907, Einstein imagined two thought experiments:

a) A person in a completely sealed box feels a force pushing them down into the boxes floor. There are two possible (but not mutually exclusive) explanations:
i. The force is due to gravity, the box is moving with constant velocity near the surface of a planet

ii. The box is in outer space, with rockets attached to it that are accelerating it “upwards”

b) A person in box feels absolutely no force at all. Again there are two explanations (these two are mutually exclusive):
i. The box is in outer space and not accelerating.

ii. The box is in free fall in a gravitational field.

Here Einstein used a bit of “classic physics reasoning” – if there is no experiment that can be done to distinguish between reasons i and ii (in either a or b), then, to a physicist, the two cases are “the same”

More generally: if two situations are experimentally indistinguishable, they are the same.

Einstein claims he came by this idea one day sitting at his desk in the patent office. Through his window, he watched a painter fall from a scaffold. The painter was briefly in free fall, leading Einstein to ponder whether there was a difference between free fall and simply being in the absence of gravity. His conclusion, expanded into the equivalence principle, was that there is not.

Strictly speaking, the thought experiments above only work for a person who is a point. If they have any height at all, and sufficiently sensitive equipment, they could measure the variation of gravity with height and tell the difference between i and ii. The difference between point and extended measurements will be discussed later in this course.

There are many ways to write “laws of physics” that fail to satisfy the equivalence principle. For example, look at Newton’s laws of motion and gravity near the surface of the Earth:

F=mia and F=mgg
mi is the “inertial mass” and mg the “gravitational mass”. There is no classical a priori reason that mi and mg should be the same. There is no requirement in any of Newton’s laws that the inertial and gravitational masses of any matter be even related to each other. Two objects could have the same inertial mass (perhaps as measured by the frequency of them vibrating on the end of a spring in deep space) and completely different gravitational masses (as measured on a balance scale perhaps).

However, many, many observations have been made, and it turns out, up to an accuracy of 1 part in 1018 that these masses are the same (or at least have a constant ratio, that could be hidden in scaling g by a constant factor).

This fact is required a priori by the equivalence principle. If the inertial and gravitational masses of objects varied, cases i and ii could be easily distinguished experimentally.