# Why does a cosmological force violate the Equivalence Principle?

Apologies if this sounds like a dumb question, I am mostly self-taught when it comes to physics.

I am trying to understand an answer I read to the question "why should we not think of the expansion of space-time as a cosmological force acting on the galaxies, pushing/pulling them apart?" The answer explained that any cosmological forces acting on comoving observers would violate the Equivalence Principle because the laws of physics must be the same in all inertial frames. The answer assumes that comoving observers are free-falling i.e there are no forces acting other than gravity, which ends up looking like a circular argument to me.

My questions are

1. Why are comoving observers in a free-fall/unaccelerated/inertial frame?
2. Why does a cosmological force that pulls everything away from everything else violate the Equivalence Principle? Isn't it doing the same thing for each comoving observer? Why/How is it a violation of the laws of physics?

There is no gravitational force causing expansion: it is simply how a uniform fluid density evolves under gravitational influence.

1) In a uniformly expanding universe, observers at rest with respect to the expansion are comoving. They are inertial because their worldlines, $$x(t) \propto a(t)$$, have no local acceleration.

2) The equivalence principal states that inertial and gravitational masses are equal; that is, a frame free-falling in a uniform gravitational field is inertial (in effect, one can "turn off" gravity by free-falling in the field). If there is an additional force than this relationship is disrupted, since free-falling under the influence of gravity is insufficient to remove all forces acting on the frame.

• I'd prefer to say "have no local proper acceleration", excellent answer though. – timm May 3 '20 at 7:34
• Thanks, but I'm still having trouble understanding. I’m going to assume that I don’t know yet whether it is really the spacetime that is expanding, or whether the spacetime is static but the galaxies are moving away due to some force. Splitting my response into multiple comments because of the text limit. – Pankaj Bhambhani May 14 '20 at 17:11
• Regarding your answer to (1), what does it mean to be at-rest w.r.t the expansion, given my assumption above? I think it would be easier to consider coming observers as those being at rest w.r.t to the matter in their vicinity. With this definition, it still possible to have coming observers in the case when there is a cosmological force acting, depending on the scale at which the force becomes dominant. – Pankaj Bhambhani May 14 '20 at 17:11
• Regarding your answer to (2), does that mean that if there were some cosmological force acting on the comoving observers then they would experience it as gravity? If yes, do we know that that’s not the case, i.e there is no “extra gravity” that appears at the largest scales? – Pankaj Bhambhani May 14 '20 at 17:12

Comoving observers are inertial because that is part of the definition of comoving coordinates.

I don't know what question or answer you read, but the cosmological constant introduces a gravitational "force" pushing everything apart. I use inverted commas because gravitational forces are inertial, or "fictional". They are the product of a coordinate choice. In comoving coordinates, inertial, comoving, observers remain stationary - there is no active force in this frame.

• If "pushing everything apart" means that the distances are increasing then the cosmological constant is not a precondition. Without a cc the universe expands decelerated which implies increasing distances too. – timm May 3 '20 at 15:36
• @timm. No, that is not what it means. Forces (even inertial forces) refer to accelerations, changes in velocites, not to changes in distances. We are not using Aristotelian physics here. – Charles Francis May 3 '20 at 17:00
• What does "pushing everything apart" mean in terms of FRW-Cosmology (which we are talking about) in your opinion? – timm May 4 '20 at 7:35
• @timm, the cosmological constant is believed to be responsible for accelerating expansion. This is not the proper place to discuss it. There are many places you can read about it. – Charles Francis May 4 '20 at 7:39
• What does "pushing everything apart" mean in terms of FRW-Cosmology (which we are talking about) in your opinion? - Note, forces would imply proper acceleration. There are no forces as things being pushed apart are in free fall. General Relativity is not about forces it is about geodesic deviation, you might want to check this article: en.wikipedia.org/wiki/Geodesic_deviation – timm May 4 '20 at 7:47