# Should the rest frame of a lab positioned on a gravitating body be considered an inertial frame in special relativity or not?

In Newtonian Mechanics, the rest frame of a lab on the Earth is considered to be (approximately) an inertial frame. The fact that a thrown ball is not moving on a straight line corresponds to the action of the force of gravity.

In special relativity, we don't have a description of gravity as a force so the movement of the ball shows that the rest frame of the lab isn't an inertial frame. However, in many gedankenexperiments we apply special relativity to situations on the (hypothetical) Earth with trains, etc. So there, the rest frame of the lab is silently taken to be an inertial frame.

This issue gets even more pronounced if we adopt the terminolgy of Taylor and Wheeler's Spacetime Physics which prefers the usuage of the term "free-falling frame" over "inertial frame" in special relativity. So they silently consider the rest frame of a lab on the Earth as "free-falling" even though it clearly is stationary relative to the Earth.

So my question is: Should the rest frame of a lab on the Earth be considered to be an inertial frame (= free-falling frame) in special relativity or not? Maybe there's something wrong with my association of the lab and "its own" inertial frame in the first place? (Taylor and Wheeler certainly suggest such an association by picturing a frame of reference as a grid of rods and clocks.)

• A free-falling frame is only subject to tidal forces. If the gravitational field varies negligibly across the space-time region of interest, one can consider the frame to be inertial and gravitation free within the framework of special relativity. – Adomas Baliuka Oct 10 '18 at 15:47

## 1 Answer

An inertial frame in SR has to be defined as a free-falling frame. There is no possible consistent framework for SR in which we would use the Newtonian definition of an inertial frame. In Newtonian mechanics, the only way we know that a frame is inertial is that we know all the forces acting on a test mass, so we can judge whether the mass is or is not obeying Newton's laws. This requires that we know, for example, all gravitational forces exerted by all objects in the universe, no matter how far away. In SR, this becomes impossible even in principle, because we can't observe things instantaneously at a distance.

However, in many gedankenexperiments we apply special relativity to situations on the (hypothetical) Earth with trains, etc. So there, the rest frame of the lab is silently taken to be an inertial frame.

Experiments of this type can be converted into equivalent experiments in a free falling frame, and nothing changes. The reason nothing changes is that, e.g., in the classic thought experiment about the train, we're only concerned with horizontal motion.

• Thanks! "Experiments of this type can be converted into equivalent experiments in a free falling frame, and nothing changes." I'm still a bit confused about terminology concerning the rest frame of the actual lab. How should we call it, if inertial isn't correct? Equivalent to an inertial frame for certain experiments? Partly inertial frame? – Marc Oct 11 '18 at 23:48