In the wikipedia page of Lorentz Transformation (http://en.wikipedia.org/wiki/Lorent...ormation_for_frames_in_standard_configuration) it's said that(adsbygoogle = window.adsbygoogle || []).push({});

The problem I'm having here is that while taking Lorentz Transformation as true, I failed to verify the statement above, i.e. I can't get " Consider two observersOandO′, each using their own Cartesian coordinate system to measure space and time intervals.Ouses (t,x,y,z) andO′ uses (t′,x′,y′,z′). Assume further that the coordinate systems are oriented so that, in 3 dimensions, thex-axis and thex′-axis are collinear, they-axis is parallel to they′-axis, and thez-axis parallel to thez′-axis. The relative velocity between the two observers isvalong the commonx-axis;OmeasuresO′to move at velocityvalong the coincidentxx′axes, whileO′measuresOto move at velocity −valong the coincidentxx′axes. Also assume that the origins of both coordinate systems are the same, that is, coincident times and positions. If all these hold, then the coordinate systems are said to be instandard configuration.O′measuresOto move at velocity −valong the coincidentxx′axes". Here's my calculation:

Is there something wrong with the calculation? Or did I just have a misunderstanding of the statement in wikipedia? Suppose that 2 frames ##O## (with observer ##A## at origin) and ##O'## (with observer ##A'## at origin) are put in standard configuration. At time ##t## in frame ##O##, ##A## measures that itself is at ##P = (x_P, 0, 0, t)## and ##A'## is at ##Q = (x_Q, 0, 0, t)## where ##x_P = 0##. Here ##P, Q## are introduced just as measurement event notation.

Now if denoted ##v = \frac{dx_Q}{dt} = \frac{x_Q}{t}## (second "=" holds because of standard configuration) and ##t' = \gamma(v) (t - \frac{v \, x_Q}{c^2}) = \gamma(v) \, t \, (1 - \frac{v^2}{t^2}) = \frac{t}{\gamma(v)}##, then ##A'## measures that ##A## is at ##P' = (x_{P'}, 0, 0, t')## and itself is at ##Q' = (x_{Q'}, 0, 0, t')## where ##\gamma(v) = \frac{1}{\sqrt{1 - v^2/c^2}}##, ##x_{P'} = \gamma(v) (x_P - v \, t) = - \gamma(v) \, v \, t## and ##x_{Q'} = 0##.

Thus the velocity of ##A## measured by ##A'## is ##v' = \frac{dx_{P'}}{dt'} = \frac{x_{P'}}{t'} = - \gamma(v)^2 \, v \neq -v##

Any help will be appreciated!

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# Are relative velocities in std config equal in value?

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