# Time-symmetry in electromagnetism: a simple puzzle

In summary, reversing the charges on two particles will have no effect on the attraction between them.
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All basic laws of physics are said to respect CPT symmetry, and Maxwell's equations in particular are time-symmetric. But here's a simple scenario I find very puzzling:

Two particles of opposite charge attract each other. In the time-reversed picture, they'd repel each other, no? But they remain opposite charges, whether under time-reversal or CPT-reversal. So it seems that the laws of electromagnetism aren't time-symmetric after all.

There has to be an obvious answer, but my aging brain won't come up with it.

They will attract each other in both cases.
In both cases, they will accelerate away from each other.
In one case, you start with them at a distance heading straight for each other - and they are accelerating as they approach each other.
In the other case, they start close to each other - but with velocity vectors pointing away from each other. They decelerate as they move away from each other.

Let's use gravity as the force. It will pull whether time goes forward or backward. Something can be in orbit and if you reverse time, it is still in orbit - but going in the same direction. In both cases, gravity is attracting them it towards the planet. Toss a rock into the air. In reverse, it bounces from the ground goes up to a highest point, and then falls into your hand.

Two particles of opposite charge attract each other. In the time-reversed picture, they'd repel each other, no?
No, the time reverse of an attractive force is still an attractive force. For example, a parabolic path pointing towards one direction is still a parabolic path pointing in the same direction when time reversed.

Thanks for the clear and timely responses -- very helpful. (Though I assume .Scott meant to say, "In both cases, they will accelerate towards each other."

Dale
It's easily seen from Newton's 2nd law (you can of course also argue within SRT): ##\vec{F}=m \vec{a}##. Since under time reversal ##t \rightarrow -t##, ##\vec{x} \rightarrow \vec{x}##, and ##m \rightarrow m##. Thus ##\vec{a}=\ddot{\vec{x}} \rightarrow \ddot{\vec{x}}##, and thus also ##\vec{F} \rightarrow \vec{F}##.

You can also argue directly with Coulomb's Law. Of course here you need in addition ##q \rightarrow +q## under time reversal. This also implies that ##\vec{E} \rightarrow \vec{E}## and ##\vec{B} \rightarrow -\vec{B}##.

## 1. What is time-symmetry in electromagnetism?

Time-symmetry in electromagnetism refers to the principle that the fundamental laws of electromagnetism remain unchanged when time is reversed. This means that the behavior of electromagnetic fields and their interactions with charged particles is the same whether time is moving forward or backward.

## 2. Why is time-symmetry important in electromagnetism?

Time-symmetry is important because it allows us to accurately predict and understand the behavior of electromagnetic phenomena. It also provides a framework for developing theories and models that accurately describe the behavior of electromagnetic fields and their interactions with matter.

## 3. What is the "simple puzzle" in time-symmetry in electromagnetism?

The "simple puzzle" in time-symmetry in electromagnetism refers to the fact that while the fundamental laws of electromagnetism are time-symmetric, some of the phenomena they describe, such as the flow of energy, are not. This poses a challenge for scientists to reconcile this discrepancy and better understand the nature of time-symmetry in electromagnetism.

## 4. How is time-symmetry in electromagnetism related to the arrow of time?

Time-symmetry in electromagnetism is closely related to the concept of the arrow of time, which refers to the one-way direction of time from the past to the future. While the fundamental laws of electromagnetism are time-symmetric, the arrow of time is evident in the fact that energy tends to flow from higher to lower concentrations and entropy tends to increase over time.

## 5. What are the implications of time-symmetry in electromagnetism for our understanding of the universe?

The concept of time-symmetry in electromagnetism has important implications for our understanding of the universe. It suggests that the laws of electromagnetism are fundamental and unchanging, and that the universe is governed by a set of timeless principles. It also challenges our understanding of time and the arrow of time, and may offer insights into the nature of the universe and its origins.

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