PlanetPhysics/Spacetime Interval Is Invariant for a Lorentz Transformation

The [[../SR/|spacetime]] interval between two events $E_{1} (x_{1}, y_{1}, z_{1}, t_{1})$ and $E_{2} (x_{2}, y_{2}, z_{2}, t_{2})$ is defined as

$(△ s)^{2} = c^{2} △ t^{2} - (△ x)^{2} - (△ y)^{2} - (△ z)^{2} .$

If $△ s$ is in [[../CosmologicalConstant/|reference frame]] $S$ , then $△ s^{'}$ is in reference frame $S^{'}$ moving at a [[../Velocity/|velocity]] $u$ along the x-axis. Therefore, to show that the spacetime interval is invariant under a [[../CosmologicalConstant/|Lorentz transformation]] we must show

$(△ s)^{2} = (△ s^{'})^{2}$

with the reference frames related by [[../LorentzTransformation/|The Lorentz transformation]] $x^{'} = \frac{x - u t}{\sqrt{1 - u^{2} / c^{2}}}$ $y^{'} = y$ $z^{'} = z$ $t^{'} = \frac{t - u x / c^{2}}{\sqrt{1 - u^{2} / c^{2}}} .$

The change in coordinates between events in the $S^{'}$ frame is then given by

$△ x^{'} = (\frac{x_{2} - u t_{2}}{\sqrt{1 - u^{2} / c^{2}}}) - (\frac{x_{1} - u t_{1}}{\sqrt{1 - u^{2} / c^{2}}}) = \frac{△ x - u △ t}{\sqrt{1 - u^{2} / c^{2}}}$

$△ y^{'} = y_{2} - y_{1} = △ y$

$△ z^{'} = z_{2} - z_{1} = △ z$

$△ t^{'} = (\frac{t_{2} - u x_{2} / c^{2}}{\sqrt{1 - u^{2} / c^{2}}}) - (\frac{t_{1} - u x_{1} / c^{2}}{\sqrt{1 - u^{2} / c^{2}}}) = \frac{△ t - u △ t}{\sqrt{1 - u^{2} / c^{2}}} .$

Squaring the terms yield

$(△ x^{'})^{2} = (\frac{△ x - u △ t}{\sqrt{1 - u^{2} / c^{2}}}) (\frac{△ x - u △ t}{\sqrt{1 - u^{2} / c^{2}}}) = \frac{(△ x)^{2} - 2 u △ x △ t + u^{2} (△ t)^{2}}{1 - u^{2} / c^{2}}$

$(△ y^{'})^{2} = (△ y)^{2}$

$(△ z^{'})^{2} = (△ z)^{2}$

$(△ t^{'})^{2} = (\frac{△ t - u △ t}{\sqrt{1 - u^{2} / c^{2}}}) (\frac{△ t - u △ t}{\sqrt{1 - u^{2} / c^{2}}}) = \frac{(△ t)^{2} - 2 u △ x △ t / c^{2} + u^{2} (△ x)^{2} / c^{4}}{1 - u^{2} / c^{2}} .$

Substituting these terms into the spacetime interval gives

$(△ s^{'})^{2} = \frac{c^{2} ((△ t)^{2} - 2 u △ x △ t / c^{2} + u^{2} (△ x)^{2} / c^{4})}{1 - u^{2} / c^{2}} - \frac{((△ x)^{2} - 2 u △ x △ t + u^{2} (△ t)^{2})}{1 - u^{2} / c^{2}} - (△ y)^{2} - (△ z)^{2} .$

Adding the first two terms with common denominators together yields

$(△ s^{'})^{2} = \frac{c^{2} (△ t^{2}) - (△ x)^{2} - u^{2} (△ t)^{2} + u^{2} (△ x)^{2} / c^{2}}{1 - u^{2} / c^{2}} - (△ y)^{2} - (△ z)^{2} .$

Pulling out a $- u^{2} / c^{2}$

$(△ s^{'})^{2} = \frac{c^{2} (△ t^{2}) - (△ x)^{2} - u^{2} / c^{2} (c^{2} (△ t)^{2} + (△ x)^{2})}{1 - u^{2} / c^{2}} - (△ y)^{2} - (△ z)^{2} .$

Factoring out a $c^{2} (△ t)^{2} - (△ x)^{2}$ in the numerator

$(△ s^{'})^{2} = \frac{(c^{2} (△ t^{2}) - (△ x)^{2}) (1 - u^{2} / c^{2})}{1 - u^{2} / c^{2}} - (△ y)^{2} - (△ z)^{2} .$

Finally, canceling terms gives

$(△ s^{'})^{2} = c^{2} (△ t^{2}) - (△ x)^{2} - (△ y)^{2} - (△ z)^{2} = (△ s)^{2} .$

Hence, the spacetime interval is invariant under a Lorentz transformation.

All Sources

^[1] ^[2] ^[3]

References

↑ Carroll, Bradley, Ostlie, Dale, An Introduction to Modern Astrophysics . Addison-Wesley Publishing Company, Reading, Massachusetts, 1996.
↑ Cheng, Ta-Pei, Relativity, Gravitation and Cosmology . Oxford University Press, Oxford, 2005.
↑ Einstein, Albert, [[../SpecialTheoryOfRelativity/|Relativity: The Special and General Theory ]]. 1916.

Template:CourseCat

[Carroll-1] Carroll, Bradley, Ostlie, Dale, An Introduction to Modern Astrophysics . Addison-Wesley Publishing Company, Reading, Massachusetts, 1996.

[Cheng-2] Cheng, Ta-Pei, Relativity, Gravitation and Cosmology . Oxford University Press, Oxford, 2005.

[Einstein-3] Einstein, Albert, [[../SpecialTheoryOfRelativity/|Relativity: The Special and General Theory ]]. 1916.

[1]

[2]

[3]

PlanetPhysics/Spacetime Interval Is Invariant for a Lorentz Transformation

All Sources

References

Navigation menu

Search