Axial symmetry, non-rotating

[PhyPsy]

I'm studying Bondi's work on gravitational radiation. He applied axial symmetry and reflection symmetry to a source of gravity. How is it that axial symmetry does not automatically imply reflection symmetry?

The assumption for axial symmetry is $\phi \rightarrow \phi'= \phi +$ constant

The assumption for reflection symmetry is $\phi \rightarrow \phi'= -\phi$

The first condition implies the second if the constant is $-2\phi$, so I don't see how the second assumption adds anything new to the source.

Also, I read that the second assumption prohibits the solution from rotating, but I don't see why this distinction needs to be made since it is invariant under any rotation.

 

[jvicens]

Hello,

Thank you for your post and for your interest in Bondi's work on gravitational radiation. I can understand your confusion regarding the relationship between axial symmetry and reflection symmetry. While it may seem that the first assumption of axial symmetry already implies the second assumption of reflection symmetry, there are some important distinctions between the two that are crucial for understanding the behavior of sources of gravity.

Firstly, it is important to note that axial symmetry refers to the symmetry of a source of gravity along its rotational axis, while reflection symmetry refers to the symmetry of the source about a plane of reflection. These two symmetries are not the same and cannot be equated.

While axial symmetry does imply that the source has a rotational axis, it does not necessarily mean that the source is symmetric about a plane of reflection. For example, a rotating disc can have axial symmetry, but it is not symmetric about a plane of reflection. This is where the second assumption of reflection symmetry comes into play. It ensures that the source is symmetric about a plane of reflection as well, which is necessary for certain solutions in Bondi's work.

Additionally, the second assumption of reflection symmetry also has implications for the behavior of the source. It prohibits the source from rotating, as you mentioned. This is because under reflection symmetry, the source must remain unchanged, meaning it cannot rotate. This distinction is important because it helps to narrow down the possible solutions for sources of gravity and allows for more accurate predictions and calculations.

I hope this helps to clarify the relationship between axial symmetry and reflection symmetry in the context of Bondi's work on gravitational radiation. If you have any further questions or concerns, please do not hesitate to ask. Thank you again for your post and for your interest in this fascinating topic.