# Einstein Equivalence Principle confusion about light beam

I am confused about one part regarding the thought experiments.

Exp 1: A lift far out in space with no gravity. the lift is just experiencing constant velocity.

Exp 2: A lift free falling in uniform gravity.

Exp 3: Lift experiences no gravity but it has an engine that accelerates upwards with a=g

Exp 4: Lift with no engine but it is stationary in a gravitational field g.

It is said the person cannot distinguish between the exp 1 and 2. But if i am in the lift and i shine a beam of light, wouldn't i be able to tell case 1 apart from case 2 since there is acceleration?

It is also said that light is bent in exp 3 and 4's case. I understand that part but not about exp 1 and 2.

In GR, it is important to distinguish between coordinate acceleration and proper acceleration. The latter is what an object physically feels and measures. On the other hand, the former can be influenced by apparent acceleration such as spacetime curvature and choice of coordinates. Objects with no source of propulsion have zero proper acceleration and follow geodesics in spacetime; they are said to be in free fall.

In experiments 1 and 2, the lift is following a geodesic. Light beams also travel along geodesics. In the frame of the lift, the four-velocity of the lift is purely in the time direction (since it is stationary), while the four-velocity of light is at a 45-degree angle (note that we are measuring vectors at the same spacetime point, since the lift and light are at the same place at the same time). Therefore, locally, light travels in a straight line in both situations because geodesics are straight. As a result, they are indistinguishable. In fact, the equivalence principle states that from a free-falling perspective, spacetime always looks like flat Minkowski spacetime locally, so experiments must give the same results.

In experiment 3, the lift is no longer following a geodesic. It has a proper acceleration of $$g$$. In its frame, geodesics will appear to have (coordinate) acceleration and curve, just like how people standing on the surface of the earth (having an upward proper acceleration of $$g$$) sees free-falling objects accelerate relative to them.

To summarize, the accelerations of the lift in experiments 2 and 3 are very different types of acceleration. The lift in experiment 2 has coordinate acceleration but zero proper acceleration, but the lift in experiment 3 has proper acceleration.

• So for exp2, a zero proper acceleration will result in same experimental results for a light beam in exp 1 and 2: light beams will not appear to be bent. Is this correct? Aug 15, 2021 at 2:25
• @Iberico Yes,that's correct. Aug 15, 2021 at 3:54

In 1. a horizontal light beam continues horizontally.

In 2. a person feels weightless, but is accelerating relative to the earth (for example). When they turn on the light, the photons accelerate downwards at the same rate - so from their point of view the beam also appears to go across the lift horizontally.

The deflection of light in a gravitational field was one of the early tests of General Relativity. It was measured when light from distant stars passed near the sun (viewed during a solar eclipse).

Deflection of light by sun

• if exp 2 works that way, wouldn't exp 3 now be also invalid in terms of light not being bent? They both have acceleration just in opposite directions Aug 13, 2021 at 16:14
• @Iberico in exp 3 the light appears to bend down ,to the observer in the lift, because the light beam really goes straight, but the floor of the lift accelerates upwards towards it. Aug 13, 2021 at 16:44

Lift-1 and 2 are not the same and you can distinguish a difference. Lift-1 has a constant force equivalent to gravity. Lift-2 has no force, and experiences weightlessness so you will definitely feel a difference. If lift-1 has a constant acceleration then eventually it will be going fast enough that you will experience horizontal light bending.