Have scientists ever experimentally proved the Andromeda paradox-like events? We all know that a light year is of the order of 1016 metres and all the objects in the space are beyond a light year.
So the Andromeda paradox should give considerable difference in time while observing an event happening even for a shorter distance like Alpha Centauri observed by a stationary observer and an observer moving (10-15 m/s) in the direction that points to the place of the event (Alpha Centauri in this case).
That being the case this paradox can be easily verified and can be used for helping people realize the effects of the non intuitive relativity.
But I haven't heard of this paradox until I dug up more deeper to know about relativity. So I was wondering if there is any problem in my thinking 
Edit: 
From what you guys say this is what I understand. Both the observers record the event (happening  at say alpha centauri) at the same time. And the time difference between the observers is not much because they are close together and the relative velocity between them is also very small compared to speed of light. While recording the event, if the moving observer understands relativity, he knows that w.r.t him  the event is happening at a time frame that is earlier to his time frame. So when he adjusts this time so that it coincides with the time frame that the stationary observer recorded the event, he will find the distance of the event to be contracted. Have I understood it right?
 A: No, this is not a thing that can be directly observed -- for several reasons.
First, in order to observe a difference, some concrete event would have to happen out at Alpha Centauri, which can be observed here with sufficient accuracy that the two observers can even form concrete impressions about when it happened. That's not a commonplace occurrence over astronomical distances.
If we have two observers with a relative speed of 100 km/h, whose time coordinates agree on Earth, their time coordinates at Alpha Centauri will differ by less than a microsecond. A comparison event in the Andromeda Galaxy (roughly a million times farther away) will still need to be pinpointable to better than about a second in order to get any clear difference. Even supernova collapses are not that quick -- neutrinos observed from the SN1987A collapse were spread out over more than ten times that.
(Hmm, there are pulsars in the Magellanic Clouds whose beeps may qualify. But if we're doing the experiment in order convince a skeptic, then we'd need to argue that the sending of a beep is an "event", which can get murky because the event is defined by the output ray from the pulsar pointing towards earth. Does that even mean the same in different coordinate systems? (Answer: Yes it does, but we'd need to provide a skeptic with an argument that makes this more certain than the common-sense conviction that simultaneity is absolute.))
Second, relativity predicts that the two observers agree that they both observed the event at the same time -- just like good old common sense says. Their only difference in opinion is that they disagree about how far away Alpha Centauri was when the event happened, and therefore compute that the event must have happened different amounts of time ago.
This means that their disagreement is not really about direct observations, but about the result of computations they make under the assumptions that relativity already works, more or less.
If we want a direct contradiction between our observers we would have to do something like radar measurements -- each observer sends a light pulse off towards Alpha Centauri; by some freak accident the two pulses arrive there at the same event and bounce back towards earth; the two observers see their pulses arrive just as they pass each other. But in order for them to draw any direct conclusions from that, they would need to have had their small mutual velocity continuously for all the eight years it took for the pulses to bounce back. Which means that both of them are flying around in spaceships rather than following Earth -- which means that this is science fiction rather than an experiment that can actually be done on these scales.
A: While I'm sure it's a translation issue, your claim that the Moon (an astronomical object) is more than a light-year away is false. As a matter of fact most of our visible Solar System is only light-minutes or light-hours distant. By the way, while is may be a cultural issue, it is never correct to claim "we all know X". The wide variety of human experience disallows it. Your post lacks specifics. Penrose's Andromeda Paradox requires non-physical people, and so it is a thought experiment which can't be executed in the real world. I also note that the GPS system uses a "correction factor" to approximate the effects of relativistic motion (gravity and speed). The fact that the time is corrected, and this allows accurate distance / location measurements proves Relativity is correct (in this context) and non-relativistic physics would fail. One recent example that may serve to answer your question is the LIGO detection of gravitational waves. The signals were detected AT DIFFERENT TIMES, which allowed their location in space to be deduced. The same effect is often observed with astronomical events, timing depends on distance. We are now able to use a clock to measure height differences of a fraction of a meter - the lower clock ticks slower. I remain confused about what it is that you are asking. Simultaneity is relative. Its well established fact. The idea that time is divided up into instants of "now" which can be shared by all observers is obviously false. I encourage you to read Penrose's book Road to Reality for more about the way the null metric (light cone) behaves in pseudo-Riemannian space-time. Taking the LIGO detectors, a clock ticked in Louisiana at the same instant that a particular peak of the signal was received. The Washington site saw that tick (from LA) at a DIFFERENT time as it observed the particular peak. Same event, different times. HTH
