John Wheeler stated "Spacetime tells matter how to move; matter tells spacetime how to curve."

Does this contradict with the assumption that mass can be accelerated in flat space-time (see i.e. this thread Acceleration in special relativity)

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    $\begingroup$ Why would it contradict? Gravity is not the only thing that can accelerate a body. There are real forces in addition to gravity. $\endgroup$ – CuriousOne Apr 1 '16 at 9:02
  • $\begingroup$ Beside gravity (mass) energy can accelerate a body. According to STR E=mc^2. So does energy curve space-time ? $\endgroup$ – ralf htp Apr 1 '16 at 9:05
  • $\begingroup$ Energy doesn't accelerate bodies. Force accelerates bodies. The formula is incomplete and as is, false. $\endgroup$ – CuriousOne Apr 1 '16 at 9:09
  • $\begingroup$ forces base on energy: difference of energy niveaus is the cause - force is the effect (i.e. free fall (potential energy)) $\endgroup$ – ralf htp Apr 1 '16 at 9:15
  • $\begingroup$ That's just one type of force. Forces are what accelerates bodies and it's forces that quantify the amount and direction of acceleration. Energy is an observer dependent quantity and always undetermined up to a constant. $\endgroup$ – CuriousOne Apr 1 '16 at 13:57

Suppose I'm orbiting the Earth. The spacetime curvature is controlling my motion i.e. I move in a circle centred on the Earth rather than a straight line because the spacetime in my vicinity is curved. This is an example of Wheeler's statement - the mass of the Earth curves spacetime and the curvature tells me how to move.

Now suppose I throw a ball I'm holding. My arm exerts a force on the ball so it accelerates and acquires a velocity relative to me. The motion of the ball is then partly due to the spacetime curvature and partly due to the force created (in some complicated way) by the actions of the cells in my arm muscles.

So there can be accelerations that aren't due to spacetime curvature. However there is an important distinction between acceleration due to an applied force and acceleration due to spacetime curvature. If I'm floating in space then I can let go of an object and it will remain floating next to me. This applies whether I'm orbiting the Earth or whether I'm floating in empty space far from any masses. My acceleration relative to a released object is called the proper acceleration and it's an important invariant in relativity. Any object that is moving solely in response to spacetime curvature has a proper acceleration of zero.

But suppose someone has sneaked up and attached a small rocket to my back and turned it on. If I now release an object I will see it accelerate away from me because I am still being accelerated by the rocket but the released object isn't. That means my proper acceleration is non-zero. A non-zero proper acceleration is what we normally think of as acceleration when doing calculations with Newtonian mechanics. The result of a non-zero proper acceleration is that we feel a force given by $F=ma$.

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  • $\begingroup$ So a difference in the niveaus of potential energy does not cause a force, because a body in free fall is weightless (force-free) ? $\endgroup$ – ralf htp Apr 1 '16 at 10:04
  • $\begingroup$ @ralfhtp: "niveaus"? Google Translate thinks it's French for "level", which doesn't help me much. $\endgroup$ – John Rennie Apr 1 '16 at 10:08
  • $\begingroup$ Yes i mean 'level'. i found the answer myself. Gravitational forces (caused by curved spacetime) are seen as fictitious forces/pseudo force. So a difference in energy levels (of potential energy) cause a ficticious force (according to GTR). On earth ground a body is more 'inertial' than in earth orbit $\endgroup$ – ralf htp Apr 1 '16 at 10:31

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