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In the principia of equivalence gravity and acceleration are the same. Wouldn't that entail that an object would require to be accelerated, or at least in movement? Sure time is passing, but wouldn't that entails that we are changing frames at an ever increase pace?

And an object to change its velocity wouldn't that require an external force? I understand the concept of non-euclidean geodesic path, but Earth and other corpuscles do not follow a circular path, more like a spiral path.

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One must distinguish between "accelerations" due to gravity, which according to the principle of equivalence are NOT accelerations at all, they are geodesics, which describe particles that move due to "gravity" and their own inertia only. There are then motions/accelerations due to external, non-gravitational forces such as electromagnetism. The problem in G.R. is that these external fields/forces are "captured" by the energy-momentum tensor, so you have the Maxwell tensor for E&M fields, and an ordinary fluid energy-momentum tensor for fluids such as the ones found in cosmology: dust, radiation, etc... This tensor exerts its influence on the spacetime curvature via Einstein's equations. So, there's no concept of forces here, just spacetime curvature. The main point of G.R., I suppose is that for partly this reason, gravity is NOT a force, it is a manifestation of spacetime curvature.

I suppose it can be summed up here in looking at the 4-acceleration of a particle, due to some 4-force. Denote the 4-velocity of some object/particle by $u^b$, then, the 4-acceleration $A^a$ would be given by:

$A^a = \dot{u}^{a} + \Gamma_{bc}^{a} u^{b} u^{c}$, where $\Gamma_{bc}^{a}$ are the Christoffel symbols obtained from the metric tensor.

Now, for geodesic motion, (just due to gravity and intertia), $A^a = 0$, so we have simply that:

$0 = \dot{u}^{a} + \Gamma_{bc}^{a} u^{b} u^{c}$,

which is the standard geodesic equation.

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In general relative it is not 3D space that is curved to form a gravitational field it is 4D space-time. It verifiably predicts that motion through time in curved space time results in acceleration in space. There can be lots of debate about what it actually means for motion in time only to result in motion through space, but there is much less doubt that the theory really works well.

Geodesics do not have to be circular, not sure why you think that. an orbital ellipse, if stable and therefore having constant energy is exactly a gravitational geodesic. If there are small energy losses, as in reality is usually the case due to tidal and solar wind effects, these will alter the course off the geodesic, producing ever smaller elliptical spirals.

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  • $\begingroup$ Thanks @JMLCarter. My question is related to acceleration and force. I have a hard time understanding how gravity is not a force, and yet is an acceleration. Again 1- If there is no need for a force, then there is a need for motion, whether motion of time or space 2- But if there is a need for motion on geodesic path, then there could be no change in velocity in other to conserve energy and no need for a force, 3- But clearly the earth does not have constant speeds nor velocity, hence my trouble. $\endgroup$ Commented Jun 6, 2017 at 15:58
  • $\begingroup$ Hmm. Suppose you move north without external force on a hypothetical surface of a spherical universe, you also move what would be described in that universe as either east or west at other points on that surface. No force, but change in direction, therefore equivalent to a force/acceleration. In GR, Gravity works like this without force. $\endgroup$
    – JMLCarter
    Commented Jun 6, 2017 at 18:07
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In a spacetime, you are always on the move with the same velocity, the speed of light. If you don't move wrt space (so only in time), this will not result in motion through space if space is flat, but it will surely do if spacetime is curved. If you find yourself freely in curved spacetime, you will always start to move in space. If you find yourself in an accelerating elevator (in outer space) it looks as if a gravity field is present (the equivalence principle). If an object is at rest initially, I think it's not hard to visualize that it starts moving to the floor of the elevator You just can't stay motionless (through space) in the frame of the elevator.

Note that to accelerate as a result of the other forces, you will have to put some effort into this because otherwise, you will not accelerate. Being accelerated by gravity is something in which you have to put no effort.

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