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I've read that gravity is time symmetric, and from what I understand, that means that if I were to record a video of objects moving only under the influence of gravity and play it in reverse, it would not go against the laws of physics. But an apple can fall down. When reversed, the apple would float upwards as though it were being repelled?

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If you toss an apple up in the air, it rises and then falls. In the time-reversed video, the apple rises and then falls.

In general, velocities change sign under time reversal, but accelerations stay the same. A downward acceleration is still downward, and a downward force is still downward, so Newton's second law is still obeyed.

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Under time reversal, $t\to-t$. Other physical quantities may change due to this transformation. For example, position does not change under time reversal. (Imagine pausing a simulation, then reversing the direction of playback. Before you begin playing back, while still paused, all objects are still where they were). On the other hand, velocity reverses direction under time reversal. (The instant you begin playing the video backwards, the objects will be moving in the opposite direction they had been been traveling.) In general, quantities depending on time with only even powers are symmetric (do not change, like position and acceleration) while quantities with odd powers are anti-symmetric (they reverse like velocity).

So in your example, first an apple drops due to gravity. Then we pause everything and apply time reversal. The apple's position is unaffected and so is its acceleration due to gravity. The apple's velocity is reversed, so it has the same magnitude but opposite direction as it had before time reversal. Then we allow everything to move again (with time reversed). The apple moves back up with its reversed velocity. That velocity changes due to acceleration so the apple slows until it reaches the point it was dropped from. The apple will exactly retrace all the points it went through when it fell with a velocity that is exactly the opposite direction as it had before. This is what is meant by saying "gravity is time symmetric".

Another question might be "why is acceleration due to gravity unchanged under time reversal"? Briefly, acceleration has even powers of time, $a \varpropto \dfrac{x}{t^2}$, so in that instant when time is reversed, acceleration does not change. Then, the force due o gravity has no time dependence, so it too is unaffected and the apple will have the same gravitational force acting on it at each position as it did when it fell.

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