Transformations in Space-Time

What are inertial systems of reference? Why are transformations between inertial systems of reference linear? A simple analysis of transformations between systems of reference.

We have two systems of reference, denoted by \mathcal{S} and \mathcal{S}'. Each system has his own co-ordinates; where (t, x, y, z) is the space-time co-ordinate for \mathcal{S} and (t', x', y', z') is the space-time co-ordinate for \mathcal{S}'. The general transformation from \mathcal{S} to \mathcal{S}' can be written as

(1) (t',x',y',z') = (t'(t,x,y,z),x'(t,x,y,z),y'(t,x,y,z),z'(t,x,y,z))

where t'(t,x,y,z), x'(t,x,y,z), y'(t,x,y,z) and z'(t,x,y,z) are unknown functions. It is convenient to use other symbols. We write t = x_0, x=x_1, y=x_2 and z=x_3. The general transformation can then be written as

(2) x'_\mu = x'_\mu(x_\nu)

A special path is known as \displaystyle \frac{d^2 x_\mu}{ds^2}=0. We call a system an inertial frame of reference, IF a free moving body will follow the path defined by \displaystyle \frac{d^2 x_\mu}{ds^2}=0. Now

(3) \displaystyle \frac{d x'_\mu}{ds} = \frac{\partial x'_\mu}{\partial x_\kappa} \frac{d x_\kappa}{ds}, thus

(4) \displaystyle \frac{d^2 x'_\mu}{ds^2} = \frac{\partial^2 x'_\mu}{\partial x_\kappa \partial x_\lambda} \frac{d x_\kappa}{ds} \frac{d x_\lambda}{ds} + \frac{\partial x'_\mu}{\partial x_\kappa} \frac{d^2 x_\kappa}{ds^2}

Now IF \mathcal{S} and \mathcal{S}' are both inertial systems of reference, then \displaystyle \frac{d^2 x'_\mu}{ds^2} = 0 and \displaystyle \frac{d^2 x_\kappa}{ds^2} = 0, therefore we obtain

(5) \displaystyle \frac{\partial^2 x'_\mu}{\partial x_\kappa \partial x_\lambda} \frac{d x_\kappa}{ds} \frac{d x_\lambda}{ds} = 0,

but if this is true for arbitrary values \displaystyle \frac{d x_\kappa}{ds} and \displaystyle \frac{d x_\lambda}{ds}, then we obtain

(6) \displaystyle \frac{\partial^2 x'_\mu}{\partial x_\kappa \partial x_\lambda} = 0

The transformation between two inertial systems of reference satisfies \displaystyle \frac{\partial^2 x'_\mu}{\partial x_\kappa \partial x_\lambda} = 0. Then it is clear that transformations between two inertial systems of reference are linear transformations. As \displaystyle \frac{\partial x'_\mu}{\partial x_\lambda} = a^\lambda_\mu, and \displaystyle \frac{\partial a^\lambda_\mu}{\partial x_\kappa}= 0, thus x'_\mu = a_\mu^\lambda x_\lambda + a_\mu. It is convenient to set a_\mu=0, then we can write x'_\mu = a_\mu^\lambda x_\lambda. Transformations between inertial systems of reference are linear transformations.

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This entry was posted in General Relativity, Physics, Relativity, Special Relativity, Transformations. Bookmark the permalink.

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