How come the concept of an instant of time is such a controversial topic with relativity, but with entanglement it's acceptable?

Question

When talking about relativity most people say that a universal instant of time doesn't exist, but when talking about entanglement most will say that the particles are both measured instantaneously. Why is talking in terms of simultaneity acceptable for entanglement but not for relativity?

Answer 1

This is an area rife with potential misunderstanding, so we need to be absolutely clear what we mean.

Suppose I take a ruler and a clock and I use rulers to mark out $x, y, z$ axes in space and the clock to note the positions of events in time. Assuming spacetime is flat, I now have a universal coordinate frame that everyone who is stationary relative to me will agree with (subject to arguments about where the origin is). Suppose I do an EPR experiment, and at some time $t$ one particle is at the point $(t, x)$ and the other is at $(t, -x)$. If Alice is sitting at the point $x$, and Bob is sitting at the point $-x$ and they both measure the state of the particle then their measurements are simultaneous i.e. they both occur at the same time $t$.

There is no problem with the use of term simultaneous as long as everyone is stationary with respect to everyone else i.e. everyone shares the same inertial frame. The same time and space coordinates apply to everyone sharing the inertial frame. To use your description, my time coordinate $t$ is universal but only everyone sharing my inertial frame.

Now suppose you are moving relative to me. Like me, you take a ruler and clock and you mark out a coordinate system that you can use for measuring the positions of spacetime points. What special relativity tells us is that your coordinates will not match mine, so if you measure the points at which Alice and Bob measure the particle state you will (in general) find they happen at different times. Depending on how you're moving you could find that Alice measures the particle state before Bob, or that Bob measures the particle state before Alice does.

This is what is meant by the failure of simultaneity. Events that are simultaneous in one frame are not necessarily simultaneous in a different frame. This applies to everything, whether you're measuring entangled states or doing any other kind of experiment.

The problem most people have is that observer $A$ could observe Alice's measurement to happen before Bob's, while observer $B$ could observer Bob's measurement to happen before Alice's. So does Alice's measurement cause Bob's result or does Bob's measurement cause Alice's result. We seem to have the paradoxical problem that both can't be true.

The solution is that neither is true. Alice's measurement does not cause Bob's result or vice versa. The two measurements are correlated, but there is no causation - only correlation.