The assumptions are,
- The two axis of the reference frames are parallel.
- The transformation between frames depends only the velocity.
- The transformation from frame to the primed frame uses a velcoity v.
- The transformation from primed frame to the frame uses a velcoity -v.
- The speed of light as measured in the two frames is constant.
- The difference in position divided by the difference in time of a light particle recorded at two positions is always the constant c.
- In the transformation, each component in the reference frame is a linear combination of the components in the primed frame.
The last condition is expressed as,
![{\displaystyle x'_{\mu }=\sum _{\nu }m_{\mu ,\nu }x_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee13f0508700cc5a9d8467b1d946ef78f526088)
where
are the components of the transformation matrix.
One approach would be two put this transformation into the Minkowski metric,
![{\displaystyle \sum _{\mu }g_{\mu ,\mu }x_{\mu }^{2}=\sum _{\mu }g_{\mu ,\mu }{x'}_{\mu }^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d03064d75e23ad4bf972cad029f250031a53bc8e)
![{\displaystyle \sum _{\mu }g_{\mu ,\mu }x_{\mu }^{2}=\sum _{\mu }g_{\mu ,\mu }(\sum _{\nu }m_{\mu ,\nu }x_{\nu })^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b7003094964d6b52996538c985551176321e69)
![{\displaystyle =\sum _{\mu }g_{\mu ,\mu }(\sum _{\nu }m_{\mu ,\nu }x_{\nu })(\sum _{\rho }m_{\mu ,\rho }x_{\rho })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82de86a5d8412f0662e355f487ed2f7057088a4)
![{\displaystyle =\sum _{\mu }\sum _{\nu }\sum _{\rho }g_{\mu ,\mu }m_{\mu ,\nu }x_{\nu }m_{\mu ,\rho }x_{\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bea85c76d56557ec3b7779c332d7dad97b89e6ab)
![{\displaystyle =\sum _{\nu }\sum _{\rho }x_{\nu }x_{\rho }\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab3a5e9b415f15f133c5dde5c81fbfec86452f)
![{\displaystyle =\sum _{\nu ,\rho }x_{\nu }x_{\rho }\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a6b8636d8ccf146fefa39ed001a26651795e71)
![{\displaystyle =(\sum _{\nu ,\nu }x_{\nu }^{2}\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }^{2})+(\sum _{\nu ,\rho }^{\nu \neq \rho }x_{\nu }x_{\rho }\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56567ec0e8c958416a8827b2bf3fab9cb6b4fe72)
Gives two equations,
![{\displaystyle \sum _{\nu }g_{\nu ,\nu }x_{\nu }^{2}=\sum _{\nu ,\nu }x_{\nu }^{2}\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13dd2849cc69f82b92e4ab78624162ae3bd8650f)
![{\displaystyle 0=\sum _{\nu ,\rho }^{\nu \neq \rho }x_{\nu }x_{\rho }\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac85f8a85631532c2f7e0a09c55fcee07656781)
From the first equation,
![{\displaystyle g_{\nu ,\nu }=\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a059b08b0f9994a9a18c46c670de44fc72fe21)
From the second equation,
![{\displaystyle 0=\sum _{\nu ,\rho }^{\nu >\rho }x_{\nu }x_{\rho }\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea6a88710e4eade2bbd9cad9dae5e50f4768254d)
![{\displaystyle \nu >\rho \implies 0=\sum _{\mu }g_{\mu ,\mu }m_{\mu ,\nu }m_{\mu ,\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b08e24b791e3b034e5c14ae2da0ea51036ca2ed)