The Brass Tacks of the Andromeda Paradox

The Subjective "Now" Interpretation of Special Relativity

Back Story

In the Andromeda paradox (aka. Rietdijk–Putnam argument), we are given to understand that there's a certain faction of hostile actors in the Andromeda galaxy who want to invade the Earth. They have presented their case to their local council and the matter has been put to a vote. An astronomer on Earth observes the verdict through a telescope and, after correcting for the propagation delay, works out the time at which our fate was sealed. A passerby sees the same image at the same time, but having a skewed perception of simultaneity, arrives at a completely different conclusion. So when exactly did the verdict make the transition from the uncertain future to the certain past? i.e. When is "now" over there?

Conventional Wisdom

Conventional wisdom says the question is invalid. What you see is what you get. All perceptions of "now" over there are real because, unlike regular spacetime events, "now" moments are subjective. But the objective reality of a spacetime event begins when light is emitted, absorbed or reflected. A "now" moment is just one for which the light (or lack thereof) has yet to be detected. If "now" moments are subjective, the content of an image in transit would be indeterminate until observed. That's certainly true from the observer's point of view, but not from that of the light source. Once the light is on its way, you can't put that horse back in the barn.

Advocates of the subjective "now" are quick to wash their hands of this issue because SR has nothing to say about the content of images. It's all about elapsed time and distances. Anyway, the traveller has a different Doppler experience than the bystander so it's easy enough to make the correction if you know who's moving and who's not. We don't though. Velocity is relative. It could well be that the Andromeda galaxy and the Earth are co-moving and the traveller is standing still. That pushes the verdict farther into the Earth's past.

In fact, everyone could be in motion. It's just a matter of perspective. We need to know the context in which we can legitimately define "now" over there by striking a perpendicular to someone's time axis.

Initial Conditions

It is demonstrably true that perceptions of simultaneity are synonymous with "now" over there for stationary or co-moving contexts. The time axis for that reference frame runs parallel to the centre of momentum. If you know who felt what forces at the launch event, conservation of momentum gives you the angle between a traveller's worldline and the centre of momentum. And that's all you need to convert a traveller's perception of "now over there" to the actual value on the centre of momentum time axis. Well that, and a practical understanding of time dilation if you want to get the right answer for high-speed cases. The point is, perceptions of simultaneity are just that - perceptions. "Now" moments are objective spacetime events like any other. Score one for the determinists.

Having said that, there is a sense in which your perceptions are real. See the Spaceship Paradox for more on that subject.

    Resources

    Here's some good resources for further study on our collective state of confusion on this topic:

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