Rotating Flat Spacetime in Minkowski Metric

[parsikoo]

In Minkowski spactime (Flat), if the coordinate system makes a rotation e.g. around y-axis (centred) , for the metric ds^2, how to make the tertad (flat spacetime) as the coordinate system rotats?

 

[haushofer]

You know how the tetrad transforms under a gct. Just apply this transformation rule for rotations. For a given metric the tetrad is only determined up to a lorentz transfo, of which spatial rotations are a subgroup. So you could start with the tetrad components being equal to 1 if you work in cartesian coordinates, and apply the transformation rule.

 

[parsikoo]

Thanks, you mean:
e(mu)=1 diagonal and for instance put e(24)=-e(42)=omega?

 

[haushofer]

I'm not sure about your notation, but writing down the transfo.law for the tetrad and parametrizing the rotation with an angle omega should do it. Just write that transfo. law here and apply the rotation.

 

[sardar]

I would like to first clarify that the term "tertad" is not a commonly used term in the field of physics. However, based on my understanding, I believe you are referring to the concept of a rotating frame of reference in flat spacetime, which is often used in the study of general relativity.

In this scenario, the Minkowski metric, which describes the geometry of flat spacetime, remains unchanged regardless of the orientation or rotation of the coordinate system. This is because the Minkowski metric is independent of any particular coordinate system and is a fundamental property of spacetime itself.

In order to make the coordinate system rotate, we can use a mathematical transformation called a Lorentz transformation. This transformation allows us to switch between different frames of reference, including rotating frames, while still maintaining the same Minkowski metric. This is similar to how a map can be rotated without changing the actual layout of the terrain it represents.

In summary, while the coordinate system may rotate, the Minkowski metric remains constant and unchanged. The use of a Lorentz transformation allows us to mathematically account for the rotation of the coordinate system and still accurately describe the geometry of flat spacetime.