At an instant, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges?

Question

Question:

In principle, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges? (At a single instance in time?)

Additional Notes:

I am aware that the evolution of this system would not behave the same because orbits cannot manifest in a simple system of like charges due to reasons regarding entropy; Just because it is entropically desirable to evolve from the big-bang state to the current universal state, it is not entropically desirable to evolve from the current universal state to the big-bang state just because we reversed the flow of time. (I intend to reverse the flow of time so that the fields reverse but not the global increase of entropy, this is why I specify At a single instance in time?).

I would ask that SE users to consider that the argument of playing a tape backwards is an ill conceived method for generating an answer since it always admits an unnatural evolution of state of the system (One that would never play out if entropy was increasing, I only reverse time for the sake of reversing the fields). For example if I tape an ink drop falling in water and watch it in reverse, it becomes immediately become apparent to me that it is being played in reverse because the system evolves in a way which violates the second law of thermodynamics. Even though this is true, what I can say with confidence is that every electric field of every particle will reverse in sign and the same can be said about the gravitational field. That is the true purpose of the question.

With that being said, I specify "(At a single instance in time?)" because I am more concerned with this idea on a fundamental level (i.e. The physical properties of the fields).

Answer 1

The temporal reverse of a system of like charges repelling each other will be a system of like charges moving toward each other, reaching a minimum radius, and then repelling again.

In particular, the sign of the force, and therefore, the acceleration is not changed under the time reversal transformation, as NowIgetToLearWhatAHeadIs' answer states.

Answer 2

No. The time reverse of a system of gravitational charges, or "point masses" to use the more usual term, is in fact a system of point masses.

To see this, imagine two masses orbiting one another under gravity. Now make a video of this and play it backwards. Now what you see is still two masses orbiting one another, just in the opposite direction. For them to be orbiting, they must be exerting an attractive inverse-square force on one another, and not a repulsive one as you supposed.

Answer 3

Suppose $\vec{r}_1(t)$ and $\vec{r}_2(t)$ are the paths of two particles interacting through a coulomb force $\vec{F}(t) = \dfrac{k q_1 q_2}{|\vec{r}_1(t)-\vec{r}_2(t)|^2} \, \, \,$, where $q_i$ is the charge of particle $i$.

Then when we play the movie backwards, we will observe the paths to be $\tilde{\vec{r}}_i(t) = \vec{r}_i(-t)$.

Differentiating once with respect to time, we will see the of the reversed motion velocity is the opposite of the velocity of the original motion as expected: $\tilde{\vec{v}}_i(t) = -\vec{v}_i(-t)$. This is because of the chain rule; differentiating with respect to time brings in a factor of $-1$.

Now to get the acceleration, we differentiate again. We get another factor of $-1$ so that the acceleration of the reversed path is the same as the acceleration of the original path: $\tilde{\vec{a}}_i(t) = \vec{a}_i(-t)$.

Thus we see that the apparent force, $m_i \tilde{\vec{a}}_i(t)$ on the particle $i$ in the reversed motion is exactly the same as the actual force in the actual motion. Therefore when you watch the movie it reverse, it does look like the particles are interacting through a coulomb force.

But let's assume for a second that the minus sign did work out so the apparent force did change sign when you reversed the movie. Then we can ask if gravity and electromagnetism are the same. Another way of saying this is does an electromagnetic interaction between opposite charges look like a gravitational interaction. The answer is that they do in the weak field limit, but at stronger fields (equivalently, large charges or masses) they don't.

Newton's inverse square law is a linear approximation made for weak fields in general relativity. If masses are sufficiently large, you will see deviations from the inverse square law, and these deviations will not be the same as the ones seen in the electromagnetic case.

In the electromagnetic case, the inverse square law again relies on the charge being small (more precisely, it relies on the fine structure constant being small). As the charges of the particles increase, the interaction between the opposite charges will start to look more and more like the interactions between quarks; that is, the attractive force between the charges is proportional to their separation instead of the inverse square of their separation. So we see for large masses and charges, the motion does not actually look the same.