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BookWei
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I read the Special Theory of Relativity in Jackson's textbook, Classical Electrodynamics 3rd edition.
Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation,
$$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$
Similarly, in the frame ##K^{'}## the wave front is specified by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=0 --- (2)$$
With the assumption that spacetime is homogeneous and isotropic, the connection between
the two sets of coordinates is linear.
The quadratic forms (1) and (2) are then related by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=(\lambda)^{2}[c^{2}t^{2}-(x^{2}+y^{2}+z^{2})]$$
where ##\lambda=\lambda(v)## is a possible change of scale between frames.
Why do we need to assume the spacetime are homogeneous and isotropic?
Will the special relativity fail if we ignore those two assumptions?
Many thanks!
Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation,
$$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$
Similarly, in the frame ##K^{'}## the wave front is specified by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=0 --- (2)$$
With the assumption that spacetime is homogeneous and isotropic, the connection between
the two sets of coordinates is linear.
The quadratic forms (1) and (2) are then related by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=(\lambda)^{2}[c^{2}t^{2}-(x^{2}+y^{2}+z^{2})]$$
where ##\lambda=\lambda(v)## is a possible change of scale between frames.
Why do we need to assume the spacetime are homogeneous and isotropic?
Will the special relativity fail if we ignore those two assumptions?
Many thanks!