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I am currently self studying general relativity and I have a question regarding the equivalence principle:

If an observer is in an elevator in deep space (or Minkowski space) where there is no gravitational force, and the elevator is accelerating upward with an acceleration of $9.8 \frac{\text{m}}{\text{s}^2}$, ignoring tidal forces, they can't distinguish between this scenario and being in an elevator resting on the surface of the earth.

How can this be possible? I mean, if I were in a spaceship that is accelerating forever at $9.8 \frac{\text{m}}{\text{s}^2}$ I'd always feel the "fictitious" force pulling me back, just like in a non-inertial frame. But if I were in the elevator at rest, I wouldn't feel any fictitious force pulling me back. Or would I?

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    $\begingroup$ "just like we feel them in non inertial frames" The acceleration of the spaceship - or, disrepectfully, the elevator in space - has exactly the same effect, locally, as gravity on Earth. $\endgroup$
    – my2cts
    Commented Apr 30, 2018 at 21:56
  • $\begingroup$ So that means that, if the elevator, or spaceship, is accelerating upwards with a=g, I can't move upward but I can move horizontally as happens on the surface of the earth. We can then recreate the earth gravitational field on our future spaceships by just accelerating them. $\endgroup$
    – AA10
    Commented Apr 30, 2018 at 22:08
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    $\begingroup$ That is exactly right, but not very practical. Rotation is a better solution. You then use the centrifugal force - another so called fictitious force. $\endgroup$
    – my2cts
    Commented Apr 30, 2018 at 22:44
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    $\begingroup$ This has to be a duplicate. You don't feel gravity. You feel every real force that acts on you except for gravity. There are roller coasters and airplanes that make you sick in the stomach because they eliminate that everything else. Gravity still acts on you almost exactly the same way it does when standing still on the ground while on zero-g roller coaster drop or in a Vomit Comet airplane ride. You don't feel gravity. There's a complex Newtonian explanation for why this is so, and a very simple relativistic explanation. The latter is that gravitation is a fictitious force. $\endgroup$ Commented May 1, 2018 at 5:13
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    $\begingroup$ If you are in an elevator at rest on Earth you very definitely feel something pulling you to the bottom of it. That's why falling over hurts, for instance. $\endgroup$
    – user107153
    Commented May 2, 2018 at 7:43

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Einstein's Equivalence principle states that freefalling reference frames are equivalent to an inertial reference frame; or to put it another way, inertial accelerative effects are equivalent to gravitational effects. This thought experiment might help:

I step into an elevator in an Earth-based building, the door closes but the elevator stays still. A varies of senses tell me which way is "up". If I jump, I quickly land back on my feet. If I try to pick up my 30kg suitcase, I can feel its weight. Now, suppose I, along with the contents of the elevator, am instantly teleported to an identical elevator in a spaceship in deep space. If the spaceship is accelerating at 1 g in a direction parallel with my conception of "up", I will have no way of knowing whether the teleportation worked and I'm on the spaceship, or it failed and I rematerialised back in the Earth-based elevator. My sense of "up", my attempts to jump or to pick up the suitcase are all indifferentiable.

So long as it's at rest in relation to the building, the Earth-based elevator's acceleration towards the centre of the Earth at 1 g is resisted by mechanical forces, which I experience as the floor exerting an upwards force on me through my feet. In the spaceship, I experience an equivalent force through my feet due to the rocket engine propelling the floor "upwards".

Now, suppose I suddenly feel weightless. My stomach lurches and I feel like vomiting. A little push of my toes sends me gently towards the ceiling. I panic and grab the suitcase handle, which slows my overall upwards trajectory but sends the suitcase towards the ceiling too, and my torso starts swinging in an arc. Did the rocket engine just cut out and I'm now in "free fall", or did the elevator cable snap and it's now falling at 9.8 $ms^{-2}$ (ignoring air resistance etc). For the moment at least, it's very hard to tell; perhaps I pick up that there's no faint vibration (the rocket thrusters), or perhaps I hear some scraping sounds (the elevator). I'll discover the truth in only a few seconds...

When I am in the elevator at rest, I don't feel any fictitious force pulling me back. Or do I?

In general relativity, gravity appears as a fictitious force; this is because GR attributes the apparent acceleration of gravity to the curvature of spacetime. You feel this fictitious force when you are in a non-inertial reference frame, e.g. "pulling you back" towards the floor in the elevator in an Earth-based building. You don't "feel" gravity in an inertial reference frame (e.g. when the elevator cable snapped) because in that frame there's no force acting on you.

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  • $\begingroup$ Ok, I think I understood. But this is true only for a certain amount of time right? In space, since the spaceship is accelerating for ever at 1g, the velocity of the ship will reach the ligh speed. If I have a clock in the ship an a clock in the elevator, the times are going to be different after a while right? $\endgroup$
    – AA10
    Commented May 2, 2018 at 10:39
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    $\begingroup$ The ship won't ever reach c, as the energy needed to accelerate at relativistic speeds becomes exponential. There are other practical but interesting issues involved in constant acceleration. And yes, the clocks will show different times due to time dilation but this starts happening as soon as the spaceship launches. $\endgroup$ Commented May 2, 2018 at 13:28
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  1. you think that your body is a particle, it is not. It is a complex system, and parts of your body have different chemical structures, and different stress-energy, and react differently on gravity and bent spacetime.

  2. you feel it because different parts of your body will be moved with different strength because of bent spacetime.

  3. you could only simply feel gravity if you could feel tidal forces from a strong gravitational field, like a black hole.

  4. when you are in freefall, you simply follow the geodesic, and so there is nothing to feel. Your body is just following curved spacetime. If you would be free falling you cannot tell if you are in space or if you fall towards Earth.

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  • $\begingroup$ The geodesics point is really cool. I have not reached that point yet in my self study. But every object that is following a geodesics path feels always weightless? If moon is following a geodesic path, doest it feel weightless? Of course Moon doesn't feel any thing, but imagine that Moon is a living being xD. $\endgroup$
    – AA10
    Commented May 2, 2018 at 10:23
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    $\begingroup$ thank you. Yes the moon would feel weightless. Just imagine, if you are orbiting the earth in a spacesuit, you feel weightless. The moon is just orbiting the Earth the same way as you (the astronaut) would orbit the Earth. $\endgroup$ Commented May 2, 2018 at 10:28

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