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This is the Lorentz transformation in 1D, x axis:

I want to get the second term of the time t equation, I mean vx/c^{2}, in two dimensions, I mean for a point in the XY plane.

I know this term arises because if we want to syncronize a point B with the origin what we do is sending a light signal from the middle point, for the frame at rest the origin is traveling to the light ray whereas point B is traveling away so the light will contact sooner the origin than the point B, that is why the clocks are not sincronized, according to the external observer.

If you have a point on the X axis you get the term v x/c^{2}, for a point on the Y axis obviously because of the symmetry light will get in both frames of reference at point B and at the origin O at the same time, syncronization will be the same. But what happens with a point on the XY plane?

I made a picture:

The middle point is A, the point we want to syncronize is B. For an external observer this systems travels to the right with speed v, and an observer will think light will get a lot sooner to point O at C, that is moving to the right, than with point B at D, because B is traveling away of the light ray. If I calculate these distances I have:

OC = v t_{1}

BD= v t_{2}

AC^{2}= AO^{2}+ OC^{2}- 2 (OC) (BD) cos (oc-bd)

AD^{2}= AB^{2}+ BD^{2}- 2 (AB) (BD) cos (ab - bd)

THen I use simply: c= space/time, to get the time:

t1= AC/c ( in this case c is the speed of light and AC is a distance from A to C)

t2= AD/c

And when I calculate:

t2 - t1 using this expressions I don't get the result I see in the books. For the the distances have square roots so I cannot simplify these expressions.

According to wikipedia these are the Lorentz transformations:

What I am doing wrong?

Thanks!

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# I Lorentz transformation in 2 dimensions

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