- #1
michael879
- 698
- 7
So the confusion I'm having here really has to do with parity inversion in spherical (or boyer-linquist) coordinates. I've been looking at the discrete symmetries of the Kerr-Newman metric, and I've noticed that depending on how you define parity-inversion, you can get very different results.
Method #1: r->r, theta->pi-theta, phi->pi+phi => x->-x, y->-y, z->-z
When you use this parity inversion, you find that C, P, and T symmetries are all independent symmetries of the metric.
Method #2: r->-r, theta->theta, phi->pi+phi => x->-x, y->-y, z->-z
Using this parity inversion, you find that the resulting metric has not transformed correctly. However if you also do a time inversion (t->-t), you find that the resulting metric describes a Kerr-Newman black hole with parameters (-M,-Q,-J). This suggests that the Kerr-Newman metric only has an MCPT symmetry, rather than independent C, P, T symmetries.
I know I must be missing something, but these two results are astoundingly different. In the first case, you just get the trivial classical result that C, P, and T symmetries all hold independently. However in the second case you find that they are not independent, that there is a new M symmetry, and that the only true symmetry is MCPT.
What am I doing wrong?
*note* M is the active gravitational mass, not the passive gravitational or inertial mass. Method #2, if true, would suggest that anti-matter has negative gravitational mass. However, this is not in conflict with either the equivalence principle or the experimental result that antimatter has positive inertial mass. Hypothetically, it would be possible, within the general relativistic framework, to have an object with gravitational mass of an opposite sign than its inertial and passive gravitational mass. Since the only anti-matter objects we observe are microscopic, there would be absolutely no way to measure its active gravitational mass.
thanks,
Mike
Method #1: r->r, theta->pi-theta, phi->pi+phi => x->-x, y->-y, z->-z
When you use this parity inversion, you find that C, P, and T symmetries are all independent symmetries of the metric.
Method #2: r->-r, theta->theta, phi->pi+phi => x->-x, y->-y, z->-z
Using this parity inversion, you find that the resulting metric has not transformed correctly. However if you also do a time inversion (t->-t), you find that the resulting metric describes a Kerr-Newman black hole with parameters (-M,-Q,-J). This suggests that the Kerr-Newman metric only has an MCPT symmetry, rather than independent C, P, T symmetries.
I know I must be missing something, but these two results are astoundingly different. In the first case, you just get the trivial classical result that C, P, and T symmetries all hold independently. However in the second case you find that they are not independent, that there is a new M symmetry, and that the only true symmetry is MCPT.
What am I doing wrong?
*note* M is the active gravitational mass, not the passive gravitational or inertial mass. Method #2, if true, would suggest that anti-matter has negative gravitational mass. However, this is not in conflict with either the equivalence principle or the experimental result that antimatter has positive inertial mass. Hypothetically, it would be possible, within the general relativistic framework, to have an object with gravitational mass of an opposite sign than its inertial and passive gravitational mass. Since the only anti-matter objects we observe are microscopic, there would be absolutely no way to measure its active gravitational mass.
thanks,
Mike