1. A Classical Analog of a Kerr Black Hole
Connor J. Hetzel
Michael J. Crescimanno, PhD (Mentor)

2. Relativistic Singularities (Black Holes)
4 exact solutions to Einstein’s equations of gravity in Minkowski space (locally flat)
Non-rotating
Rotating
Uncharged
Schwarzschild
Kerr
Charged
Reisnner–Nordström
Kerr–Newman

3. Kerr Metric Black Hole 4 constants of motion for particles in orbit
Energy
Axial angular momentum
Rest mass
Carter-constant
Clifford Will found Newtonian systems that exhibit a “Carter-like” constant of motion
Found prolate mass distribution
Proposed search for oblate case
Clifford Will’s paper: Carter-like constants of the motion in Newtonian gravity and electrodynamics

4. Multipole Expansions Convenient method of accounting for mass
Gravitational and electrodynamic slightly different
Will proposes elucidation of nature of Carter constants

5. Will’s Conditions
Will gave conditions that force a “Carter-like” constant
Q 2l+1 = 0
Q 2l = m (Q 2 / m) l
Where Q l are the gravitational multipole moments
Conveniently very similar to Kerr
Q l + iJl = m(ia) l
Q 0 = mass
a = normalized angular momentum

6. Our Approach Find oblate solutions to Will’s question Q 2 < 0
Compare these classical analogs to properties of Kerr metric
Angular momentum bound in GR
No bound in Newtonian gravity

7. Solutions Found Case 1: ρ = Σeven Bl r β(l)e - λr Pl (z)
Bl+2 = Bl (Q 2 λ2 / m) (2l+5)/(2l+1)/(l+4+β(l+2))/(l+3+β(l+2))
Pl is the usual Legendre Polynomial of order l
Case 6: ρ = Σeven Bl e -a(l)Pl (z)
a(l) = dr n+3/ λn+3
Bl+2 = Bl (Q 2 /m λ2)(l+5)(2l+5)/(n+3)/(2l+1)
Other 4 cases well understood by previous work

8. Goals Case 1 Replicate old limiting conditions
Evaluate positivity of energy
Examine shape of naive horizon
Case 6
Understand issues with convergence

9. Results Case 1: Issues replicating old convergence (different value)
Single-value issues

10. Results Problem with old work:
‘Simplifying’ cases 1 & 3 made density f(θ) at origin
Cases 2, 5, & 6 always f(θ) at origin
Case 4 had no single-valued issues
Non-single-valued ⇒ density functions unphysical

11. Future Work Re-evaluate cases 1 & 3 β(l) ≠ 0 ∀l
Potentially re-evaluate cases 2, 5, & 6
Present form cannot force single-valued density at origin
Evaluate other physicality constraints
Positive energy
Closure of horizon (causality of spacetime)

12. Acknowledgements Department of Physics and Astronomy
YSU Choose Ohio First Scholars Program
Case Western Reserve University Polymer REU

13. Questions?