I was reading about Andromeda paradox, and I started wondering. How can we know that the situation in the Andromeda paradox is real ? How can you know that simultaneity is real ? How can you say that you are simultaneous with a star fleet coming from Andromeda, if there is no way of knowing it ? For simultaneity to be meaningful, shouldn't it propagate at the speed of light ?
So, is simultaneity testable ?
Simultaneity is real in precisely the same sense that the component values of a vector or tensor are real: your instruments will certainly measure certain things as co-incident and those measurements are real and instruments can measure vector components. But these events are not needfully co-incident in other inertial frames, just as the component values of a vector will be transformed. However, the descriptions from different inertial frames are equivalent - they convey the same information and are linked, in SR, by the information-preserving (i.e. bijective) Lorentz transformation.
How can you say that you are simultaneous with a star fleet coming from Andromeda, if there is no way of knowing it ? For simultaneity to be meaningful, shouldn't it propagate at the speed of light ?
For events in Andromeda - or other far off events, you can only declare their simultaneity relative to your inertial frame of reference after "betokening" or "announcing" information reaches you. So, from your inertial frame, you can only conclude that a radioactive decay at your position was simultaneous with some event in Andromeda "announced" to you by the arrival of e.g. radio signals 2.538 million years afterwards. So in this case, your declaration of simultaneity of the decay and Andromedean event can only be made 2.538 million years after you observed the radioactive decay.
The Rietdijk-Putnam argument (I wasn't aware of Rietdijk - I knew Putnam argued this way) is simply making a case for eternalism: it is most decidedly not a paradox in the strict sense of being a logical contradiction. It's simply saying that in the light of relativity, the reality of an event or thing cannot be restricted to a single instant in time: everything must be thought to be a four dimensional object with a real, nonzero extent in spacetime for special relativity to avoid logical contradictions. Other names for someone who subscribes to this kind of philisophical position are Detenser, Eternalist, Perdurist and Four-Dimensionalist. I should declare my own position as being decidedly eternalist.
Your question and line of thought are very keen and insightful because they allow you insight into why simultaneity can be frame dependent and yet not beget logical contradictions. It is because there is no way of instantaneously communicating or comparing experimental results between distant observers. When one takes account of the lightspeed limit in bringing together experimental results for comparison, it is shown that no violations of causality are possible: even though relatively moving inertial observers may disagree on the time between two events at the same spatial point, they will not disagree on the sign - if one event happens before the other in one frame, it must do so in the other. Be careful here: in general Lorentz transformations do indeed alter the order of events, but they do not re-order events at the same point (see footnote). If event $A$ can be causal of event $B$, all inertial observers will deduce this relationship.
Footnote: A word of warning: you may see some authors (although not ones talking about SR alone) talk about Lorentz transformation as meaning any transformation from the group $O(1,3)$. This includes transformations that invert order of events even at the same spatial point. A subgroup of $O(1,3)$ is the topologically connected component $SO(1,3)$ or $O^+(1,3)$ containing the identity. This contains only orthochronous ones - ones that do preserve order at the same spatial point and it is this one that we talk about in special relativity. I've seen many Wikipedia pages use the more general usage, although I don't think any talking about SR do this.
An observer at any given location can only define events to be simultaneous in retrospect, after there's been time for the light from each event to reach them. Simultaneity in each inertial frame is defined based on the assumption that light travels at a uniform speed in that frame (the second postulate of special relativity), so for example if at the stroke of midnight on Jan 1. 2010 I receive light from an event that happened exactly 10 light-years away in the coordinates of my rest frame, and at midnight Jan 1. 2020 I receive light from an event exactly 20 light-years away, I conclude that in my frame both events happened simultaneously at midnight on Jan. 1 2000.
The fact that light travels at the same coordinate speed in each inertial frame, which implies that different frames must give different answers to whether two events share a common time-coordinate (see this video about Einstein's train/lightning thought experiment, perhaps along with my answer here which discusses it, if you don't see why the first statement implies the second), is really just a matter of definition--the Lorentz transformation which relates the coordinates of one frame to the coordinates of another guarantees that these things are true. The physical prediction of relativity is that once you have constructed such a set of coordinate systems, you will find that the laws of physics obey the same equations when expressed in terms of any system's position and time coordinates--that the laws of physics are "Lorentz-invariant". Since the laws of physics obey exactly the same equations in all these different frames, and the frames disagree about simultaneity, there is no basis in the laws of physics for "preferring" one frame's definition of simultaneity over any other's. But that's an experimentally falsifiable claim since we might potentially someday discover a new fundamental law of physics that doesn't have this property of Lorentz-invariance, and if its equations looked simplest (or had some other 'nice' property) when expressed in the position and time coordinates of one unique frame, then we could say that frame was a "preferred" one and also treat its definition of simultaneity as preferred.
It's a paradox about relativity.
How can you know that simultaneity is real?
To keep things simple, we can define simultaneity of two events for a particular observer as light (or information) reaching the observer at the same time. According to this definition, simultaneity is a "real" phenomenon. A layer of complexity (that doesn't much affect the paradox) can be added by having the observer discount the time required for light or information to reach the observer to determine simultaneity.
For simultaneity to be meaningful, shouldn't it propagate at the speed of light?
As defined above, it's based on information that propagates at the speed of light. The simultaneity itself doesn't need propagate. Knowing all relevant trajectories, any observer can directly calculate the time difference between events for any other observer. Thus its possible to know that a different observer will perceive two events as simultaneous before information from that observer's actual detection of the events reaches us.
The "paradox" is that special relativity is weird and counter-intuitive. And it is.
