Lorentz Transformation: Conservation of Inner Product?

[princeton118]

Is Lorentz transformation one kind of coordinate transformation? If so, the conservation of inner product, which is the primary property of Lorentz Transformation, is trivial, isn't it? The vector transforms in some way, the metric transforms in the inverse way, so the inner product defintely will not change under a coordinate transformation. If so, any coordinate transformation can be Lorentz transformation?

 

[robphy]

The transformation must be linear as well.

 

[Entropia]

Yes, Lorentz transformation is a type of coordinate transformation that describes how measurements of space and time change between different inertial frames of reference. The conservation of inner product is a fundamental property of Lorentz transformation, as it ensures that the laws of physics remain the same in all inertial frames.

While it is true that the inner product will not change under a coordinate transformation, not all coordinate transformations are Lorentz transformations. Lorentz transformation specifically refers to the transformations that preserve the speed of light, which is a fundamental constant in physics. So, while any coordinate transformation may preserve the inner product, not all of them will preserve the speed of light and therefore cannot be considered Lorentz transformations.

In summary, Lorentz transformation is a specific type of coordinate transformation that preserves the speed of light and the conservation of inner product is a key property of this transformation. While any coordinate transformation may preserve the inner product, not all of them will preserve the speed of light and thus cannot be considered Lorentz transformations.