Physical Meaning of r in BL Coordinates

[marschmellow]

Not much else to say other than the title. In the Schwarzschild spacetime, the radial coordinate r didn't represent radial distance, but it at least represented the thing that determines the area of a sphere centered on the large mass. It doesn't seem like that interpretation can be given to the Boyer-Lindquist r, but can any other physical interpretation be given to it?

Thanks in advance.

 

[Bill_K]

marschmellow, Yes, the radial coordinate r used in the Boyer-Lindquist version of the Kerr metric has an important physical meaning, but it's not as easy to visualize geometrically as the Schwarzschild coordinate r. The reason is that Kerr (and also Kerr-Newman and Kerr-NUT) are best understood as complexified generalizations of Schwarzschild. Consequently, the geometric meaning of r is also complexified.

Note the factor r² + a² cos² θ that appears in the metric in several places? (It's often abbreviated ∑.) This factor is really the norm of a complex radial quantity, ρ ≡ r + ia cos θ.

The geometrical significance of ρ is tied most directly to the Weyl tensor rather than the metric. Both Schwarzschild and Kerr are examples of what are called Type D metrics. Which means that in an appropriately chosen frame, only one component of the Weyl tensor survives. For Schwarzschild, the Weyl tensor falls off cubically with distance, Ψ₂ = M/r³. The "geometrical significance" of r is therefore, r = constant are the surfaces on which the spacetime curvature is constant. For Kerr, Ψ₂ = m/ρ³. The last expression is complex, so actually there are two curvature components, the real and imaginary parts of Ψ₂.

 

[marschmellow]

marschmellow, Yes, the radial coordinate r used in the Boyer-Lindquist version of the Kerr metric has an important physical meaning, but it's not as easy to visualize geometrically as the Schwarzschild coordinate r. The reason is that Kerr (and also Kerr-Newman and Kerr-NUT) are best understood as complexified generalizations of Schwarzschild. Consequently, the geometric meaning of r is also complexified.

Note the factor r² + a² cos² θ that appears in the metric in several places? (It's often abbreviated ∑.) This factor is really the norm of a complex radial quantity, ρ ≡ r + ia cos θ.

The geometrical significance of ρ is tied most directly to the Weyl tensor rather than the metric. Both Schwarzschild and Kerr are examples of what are called Type D metrics. Which means that in an appropriately chosen frame, only one component of the Weyl tensor survives. For Schwarzschild, the Weyl tensor falls off cubically with distance, Ψ₂ = M/r³. The "geometrical significance" of r is therefore, r = constant are the surfaces on which the spacetime curvature is constant. For Kerr, Ψ₂ = m/ρ³. The last expression is complex, so actually there are two curvature components, the real and imaginary parts of Ψ₂.

Great answer. Thank you so much!