Compatible, incompatible, possible, impossible

I wish to clarify the four concepts of the title, which are sometimes confused when talking about physical theories and their application:

• An event (real or imagined) can be compatible with a theory. In this case, if the event were real, it would not pose any problem for the theory, which admits in principle the possibility of that event taking place.
• An event (real or imagined) may be incompatible with a theory. If the event is real, its existence defies the theory. If the theory fails to explain it, the theory should be improved or replaced by another. Something like this happened when Newton's theory of Universal Gravitation was replaced and improved by Einstein's General Relativity. If the event is imagined (such as the EPR paradox, proposed by Einstein, Podolsky, and Rosen to combat quantum mechanics), the challenged theory should be defended by showing that the imagined event either does not occur in reality, or if it does, it fits the predictions of the theory.
• Two theories are incompatible, if there is at least one phenomenon or event incompatible with one of them, but not with the other. Otherwise, they are compatible.
• An event (real or imagined) can be possible or impossible. A possible event is one that can happen. If it is impossible, it cannot happen under any circumstances. This is so, whatever the specific theory with which at a given moment we are trying to explain the phenomena we perceive. We know, for example, that squaring the circle is impossible because π is a transcendental number. This happens independently of our physical theories, present, past or future.

The two alternatives (compatible or incompatible, possible or impossible) are independent of each other. A possible event may be incompatible with a theory; an impossible event may be compatible with it. Let's look at a few examples:

• The theories of general relativity and quantum mechanics are incompatible. The first describes better those phenomena that have to do with the movements of heavenly bodies. The second, what happens in the world of atoms and subatomic particles. But each of the two theories fails when it is used in the field of application of the other. Therefore, neither of these two theories can be considered final. Current theoretical physics is incomplete, because it is based on two incompatible theories.
• Events such as time travel or travel at superluminal speeds are probably impossible, because they give rise to destructive paradoxes that, if they happened, would make the universe behave absurdly. In both cases it is usually said that these travels could be compatible with the theory of relativity in certain circumstances. For instance, for the second type of travel, it’s usual to speak about the warp drive, based on the Alcubierre metric. But this means nothing, if the phenomena in question are impossible. In case of discrepancy, the theory must yield priority to the phenomena it tries to explain.

In a previous post I mentioned the paradoxes associated with time travel. In this post I described in detail an equally destructive paradox that would take place if it were possible to send messages from one place to another, faster than the speed of light. If travel at superluminal speeds were possible, it is obvious that information could automatically be sent at the same speed, so the paradox could happen.

Karl Popper

To say that an event is compatible with a theory does not mean that it is possible, because theories change and can never be considered fully confirmed, as Karl Popper rightly pointed out.

Saying that the paradoxes can be resolved if at the end of the trip we find ourselves in another universe does not solve anything, because we are speaking about traveling in time or space without leaving our universe.

The same post in Spanish

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Manuel Alfonseca