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I was going through special relativity and got stuck with few questions:

Is Lorentz transformation just the co-ordinate transformation, or is the space actually changing during the transformation?

I have always thought of it as just a coordinate transformation. When I consider that, the time dilation effect we see after the transformation seems like just a consequence of mathematics rather than a real physical event(process) we get (due to the change in our way of looking at the same space). If it is true then I can't come to the conclusion of how time dilation is an actual physical process?

If Lorentz transformation is the transformation of space, then why is it so? What is special about it that makes it so? 

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  • $\begingroup$ Could you explain the difference between a coordinate transformation and an actual space transformation? For example, is there any possible experiment (whether technically feasible or not) that would give one result if it is a coordinate transformation and a different result if it is an actual space transformation? $\endgroup$
    – Dale
    May 6, 2020 at 0:17
  • $\begingroup$ Welcome New contributor user263390! Aren't the coordinates in some inertial frame (usually) the time as read by (synchronized) clocks and rulers at rest with respect to each other in that frame? $\endgroup$ May 6, 2020 at 0:54

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Consider a coordinate system with Cartesian coordinates and an origin. Say you are sitting on the y axis at y = 1 and your friend is sitting at the x axis at x = 1. Now, say you both define your own positions as the new origin and label the coordinates of every other point in space using the Cartesian coordinate system $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$. Would you both have the same labels for each point? No, because your labels depend on what you chose as origin. There is however a transformation $T$ that can be performed that could map each point in space from your coordinate system to your friend's, $$(x_1,y_1,z_1)\rightarrow (x_2,y_2,z_2)$$ I assume this is what you mean by coordinate transformation. Now, consider that you and your friend are moving at different constant speeds. If you both measure the speed of a third object, you would get different results. But for non-relativistic speeds, you can transform the coordinates between your and your friend's, using Galilean relativity. Now, if the speeds in question are near light speed, the Galilean relativity breaks down and you need to use Lorentz transformation to map points between coordinate systems. In addition to measuring different speeds, you also measure different time and length now. So, Lorentz transformation is just a coordinate transformation between inertial frames of reference (traveling at constant velocity). The physical effects observed due to the coordinate transformation are only true for the observing frame of reference. They are not the same in another frame, moving with a different speed. So it is an effect of the coordinate system we choose to observe from.

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