How Faster than Light Speed Breaks CAUSALITY and creates Paradoxes
If you point a powerful laser at the moon, and spin it 100 times per second, the dot on the moon will move 3X the speed of light. This is ok. The maximum speed limit is not a limit with which things can move, but is a limit on the speed of causality. A cause cannot have an effect anywhere in the universe faster than the speed of light.
What are the implications of having a speed limit on causality? Why is there a limit in the first place? And how would causality be broken if information could travel faster than light.
Only a force can cause something. The speed of information is the speed of a force field. You can’t send information faster than the force field can change. This sets an upper limit on its speed. Light in a vacuum travels at c, the maximum speed because photons are massless. Without mass there is no restriction on its velocity.
Einstein’s theory of special relativity is based on two postulates. Neither of the postulates state that FTL is impossible. Postulate 1: The laws of physics are the same in every reference frame. Postulate 2: The speed of light in a vacuum is constant, and independent of the motion of the source of that light.
But what if this maximum speed was infinite, implying that the speed of light is infinite? For one thing using the equation E=MC^2, it would require an infinite amount of energy to accumulate any mass in the universe. So no massive particles could form.
The other problem is that we would not have light at all. In Maxwell’s equations. c equals one over the square root of the permittivity and permeability of free space. Permittivity is the resistance of free space to the formation of electric fields, and permeability is the formation of a magnetic field by an electric current. But if you set c equal to infinity, it would mean that these interactions of electricity and magnetism would not happen. There would be no waves, and thus, no light.
What if we kept special relativity, but we allowed faster than light communication, for example like instant subspace communication like in Star Trek? The problem is that in special relativity, it is not clear what instantaneous means. What’s happening “right now” depends on how fast, and in which direction you’re moving. There is no absolute now. The now depends on the reference frame.
For example, imagine Alice on earth, and Bob in a rocket traveling 0.87c towards Proxima Centauri, our nearest neighboring star.
The world line for Alice will be straight up, because she will not be moving in her frame of reference, but she will be moving forward in time. Her clock on her world line will tick normally for her. Bob’s clock will be moving at half the rate of Alice’s due to Bob’s speed relative to Alice
But from Bob’s perspective, he is not moving, and the earth is moving away from him at 0.87c. So from Bob’s perspective his clock is running normally. And it is Alice’s clock that is running at ½ the speed of Bob’s clock.
Here’s how this would break causality: Alice sends a message to Bob using an instantaneous signal to Bob at her 4 seconds. It would arrive to Bob when his clock reads 2 seconds. But from Bob’s perspective, Bob receives the signal at 2 seconds, when Alice’s clock was at 4 seconds. But her clock is running slower than Bob’s from Bob’s perspective. This means that the signal from Alice would have to travel back in time to reach Bob.
Because Bob receives the signal at 2 seconds, he could read the signal and reply at his 4 seconds using the same instantaneous “subspace” signal. That means Alice would receive the reply at her 2 seconds. She would receive the reply BEFORE she sent the message at 4 seconds.
If this was permissible, Alice could use Bob to send messages to herself in the past. She could send a message to herself at 2 seconds, to not the send the message to BOB at 4 seconds. In that case, how did she receive a reply to a message that she never sent?
Anything that allows information to travel to the past causes paradoxes like this and the grandfather paradox, among others. So such a thing is not permissible, and does not happen according to physics as we understand it. There is no absolute time. And there is no absolute now.
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