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[...] as defined by Einstein in his book "Relativity the special and general theory") it [simultaneity] is defined as a matter of observation. Einstein's literal prescription can be read for instance here: http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I#Section_8_-_On_the_Idea_of_Time_in_Physics To summarize in my own ...

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First, to be clear, events are points in spacetime that exist independent of any coordinate system as is the interval associated with two events. So, we can say, without introducing a coordinate system, that the interval associated with two events is timelike, lightlike, or spacelike In the timelike case, we can say that one event is later than the other, ...

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Simultaneity is a convention in special relativity, not an observation. The Einstein clock synchronization procedure is one way of defining a plane of simultaneity in space-time; however, special relativity can be (and has been by, e.g., John Winnie in 1970) reformulated with a wide range of allowable clock synchronization schemes with no effect on the ...

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Simultaneity depends on the frame of reference. The statement that two events are simultaneous in a reference frame is just the statement that in the time coordinate of the two events in that frame are the same. This definition in general depends on the reference frame. However, there are two exceptions where whether or not the events are simultaneous is ...

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Disclaimer: I've never read this particular book, but I'm reasonably certain index notation is invariant across most of physics. The mathematical notion of interest is of course the inner product in some, well, inner product space. The definition of an inner product is given by a bilinear form - the metric. Let's call it $g$. Then $$(a, b) \equiv g(a, ... 2 The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance. To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just:$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2  Suppose you want to ...

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I'm not an expert on this particular topic, but I believe I can answer your question. There are different kinds of "dimensions". The standard 3 spatial dimensions we live in are infinite in extent. However, one can also imagine dimensions that have a periodicity (like a circle). In such cases, there is a "size" to the dimension that refers to the ...

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