Collision kinematics of charged test particles at arbitrarylatitude near degenerate horizons

Filip Hejda

Centro Multidisciplinar de Astrofísica, Departamento de Física, Instituto Superior Técnico,Universidade de Lisboa

in cooperation with:M. Kimura, O. B. Zaslavskii, J. Bičák, J. P. S. Lemos

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 1 / 9

Introduction

History of black holes as particle accelerators/colliders:

R. Penrose, Gravitational Collapse: the Role of General Relativity,Rivista del Nuovo Cimento, Numero Speziale, 252 (1969).

J. Bardeen, W. H. Press, S. A. Teukolsky, Rotating black holes:Locally nonrotating frames, energy extraction, and scalar synchrotronradiation, The Astrophysical Journal 178, 347-370 (1972).

T. Piran, J. Shaham, J. Katz, High efficiency of the Penrosemechanism for particle collisions, The Astrophysical Journal Letters196, L107-L108 (1975).

M. Banados, J. Silk, S. M. West, Kerr Black Holes as ParticleAccelerators to Arbitrarily High Energy, Physical Review Letters 103,111102 (2009).

. . . and then dozens of others. . .(including F.H., J. Bičák, arXiv:1612.04959, PhysRevD.95.084055)

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 2 / 9

Introduction

History of black holes as particle accelerators/colliders:

R. Penrose, Gravitational Collapse: the Role of General Relativity,Rivista del Nuovo Cimento, Numero Speziale, 252 (1969).

J. Bardeen, W. H. Press, S. A. Teukolsky, Rotating black holes:Locally nonrotating frames, energy extraction, and scalar synchrotronradiation, The Astrophysical Journal 178, 347-370 (1972).

T. Piran, J. Shaham, J. Katz, High efficiency of the Penrosemechanism for particle collisions, The Astrophysical Journal Letters196, L107-L108 (1975).

M. Banados, J. Silk, S. M. West, Kerr Black Holes as ParticleAccelerators to Arbitrarily High Energy, Physical Review Letters 103,111102 (2009).

. . . and then dozens of others. . .(including F.H., J. Bičák, arXiv:1612.04959, PhysRevD.95.084055)

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 2 / 9

Introduction

History of black holes as particle accelerators/colliders:

R. Penrose, Gravitational Collapse: the Role of General Relativity,Rivista del Nuovo Cimento, Numero Speziale, 252 (1969).

J. Bardeen, W. H. Press, S. A. Teukolsky, Rotating black holes:Locally nonrotating frames, energy extraction, and scalar synchrotronradiation, The Astrophysical Journal 178, 347-370 (1972).

T. Piran, J. Shaham, J. Katz, High efficiency of the Penrosemechanism for particle collisions, The Astrophysical Journal Letters196, L107-L108 (1975).

M. Banados, J. Silk, S. M. West, Kerr Black Holes as ParticleAccelerators to Arbitrarily High Energy, Physical Review Letters 103,111102 (2009).

. . . and then dozens of others. . .(including F.H., J. Bičák, arXiv:1612.04959, PhysRevD.95.084055)

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 2 / 9

Introduction

J. Bardeen, W. H. Press, S. A. Teukolsky, Rotating black holes:Locally nonrotating frames, energy extraction, and scalar synchrotronradiation, The Astrophysical Journal 178, 347-370 (1972).

M. Banados, J. Silk, S. M. West, Kerr Black Holes as ParticleAccelerators to Arbitrarily High Energy, Physical Review Letters 103,111102 (2009).

. . . and then dozens of others. . .(including F.H., J. Bičák, arXiv:1612.04959, PhysRevD.95.084055)

Where does the energy come from?

Is it possible to remain stationary at the radius of the horizon of ablack hole? Well, it shouldn’t be.

However, first, there exists a type of motion which asymptoticallyapproaches the radius of an orbit

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 2 / 9

Introduction

J. Bardeen, W. H. Press, S. A. Teukolsky, Rotating black holes:Locally nonrotating frames, energy extraction, and scalar synchrotronradiation, The Astrophysical Journal 178, 347-370 (1972).M. Banados, J. Silk, S. M. West, Kerr Black Holes as ParticleAccelerators to Arbitrarily High Energy, Physical Review Letters 103,111102 (2009).. . . and then dozens of others. . .(including F.H., J. Bičák, arXiv:1612.04959, PhysRevD.95.084055)

Where does the energy come from?Is it possible to remain stationary at the radius of the horizon of ablack hole? Well, it shouldn’t be.However, first, there exists a type of motion which asymptoticallyapproaches the radius of an orbitSecond, in the limit of extremal Kerr, some orbits are on the horizonradius. That allows for divergent redshift factor.Let us examine how this unphysical (requires extremality andfine-tuning), yet interesting phenomenon behaves at general latitudeof the horizon of an electrovacuum black hole

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 2 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

An electrovacuum black hole

g = −N2 dt2 + gϕϕ (dϕ− ω dt)2 + grr dr2 + gϑϑ dϑ2

A = At dt + Aϕ dϕ = −φdt + Aϕ (dϕ− ω dt)

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Motion necessarily forbidden if

|uK|2 > −1

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

For critical particles (with XH = 0)

|uK|2 = l2 |dϕ|2 + q2∣∣∣A∣∣∣2 − 2l q dϕ · g−1 · A

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

An electrovacuum black hole

g = −N2 dt2 + gϕϕ (dϕ− ω dt)2 + grr dr2 + gϑϑ dϑ2

A = At dt + Aϕ dϕ = −φdt + Aϕ (dϕ− ω dt)

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Motion necessarily forbidden if

|uK|2 > −1

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

For critical particles (with XH = 0)

|uK|2 = l2 |dϕ|2 + q2∣∣∣A∣∣∣2 − 2l q dϕ · g−1 · A

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

An electrovacuum black hole

g = −N2 dt2 + gϕϕ (dϕ− ω dt)2 + grr dr2 + gϑϑ dϑ2

A = At dt + Aϕ dϕ = −φdt + Aϕ (dϕ− ω dt)

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

For critical particles (with XH = 0)

|uK|2 = l2 |dϕ|2 + q2∣∣∣A∣∣∣2 − 2l q dϕ · g−1 · A

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

For critical particles (with XH = 0)

|uK|2 = l2 |dϕ|2 + q2∣∣∣A∣∣∣2 − 2l q dϕ · g−1 · A

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

For critical particles (with XH = 0)

|uK|2 = l2 |dϕ|2 + q2∣∣∣A∣∣∣2 − 2l q dϕ · g−1 · A

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Electrogeodesic test particle motion Border of “allowed phase space”

Decomposition of velocity of a charged test particle

u = uK + unK uK = g−1 · (−εdt + l dϕ− qA)

Normalisation of velocity

|u |2 = |uK|2 + |unK|2 = −1 |unK|2 > 0

Boundary of “allowed phase space”

|uK|2 = −1

Pass to near-horizon azimuth and electromagnetic potential

dϕ = dϕ− ωH dt A = A+ φH dt

Rearrangement of constants of motion

uK = g−1 ·(−XH dt + l dϕ− qA

)X ≡ ε− lω − qφ

We get a hyperbola. What does it tell us?

−l2 |dϕ|2 − q2∣∣∣A∣∣∣2 + 2l q dϕ · g−1 · A = 1

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 3 / 9

Results for extremal Kerr-Newman solution Example of the orientation of the hyperbola

aM =

√5−√172 , ϑ = 1

2 arccos√17−52

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9

l M

q

εcr =2

εcr =3

|uK|2 = −1critical energy lines

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 4 / 9

Results for extremal Kerr-Newman solution Space of hyperbolas

Kerr Reissner-Nordström

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ϑ

γKN (a = M cos γKN,Q = sin γKN)

1a2b 1b2a

1b2b

1b2b

1a2a

1c2c

1b2c31a2c3

3

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 5 / 9

Magnetised Kerr-Newman solution

How can we charge a black hole? In external magnetic field, there is Waldcharge accretion:

R. M. Wald, Black hole in a uniform magnetic field, Phys. Rev. D 10,1680-1685 (1974).

Using solution generating techniques, one can transform Kerr-Newmansolution into magnetised Kerr-Newman (MKN) solution, which:

describes charged, rotating black hole in a strong external magneticfield

is not asymptotically flat (one would say it’s asymptoticallypathologic)

behaves like strong-field end-state of Wald’s charge accretion, see:W. A. Hiscock, On black holes in magnetic universes, J. Math. Phys.22, 1828-1833 (1981).

is complicated

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 6 / 9

Magnetised Kerr-Newman solution

How can we charge a black hole? In external magnetic field, there is Waldcharge accretion:

R. M. Wald, Black hole in a uniform magnetic field, Phys. Rev. D 10,1680-1685 (1974).

Using solution generating techniques, one can transform Kerr-Newmansolution into magnetised Kerr-Newman (MKN) solution, which:

describes charged, rotating black hole in a strong external magneticfield

is not asymptotically flat (one would say it’s asymptoticallypathologic)

behaves like strong-field end-state of Wald’s charge accretion, see:W. A. Hiscock, On black holes in magnetic universes, J. Math. Phys.22, 1828-1833 (1981).

is complicated

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 6 / 9

Magnetised Kerr-Newman solution

How can we charge a black hole? In external magnetic field, there is Waldcharge accretion:

R. M. Wald, Black hole in a uniform magnetic field, Phys. Rev. D 10,1680-1685 (1974).

Using solution generating techniques, one can transform Kerr-Newmansolution into magnetised Kerr-Newman (MKN) solution, which:

describes charged, rotating black hole in a strong external magneticfield

is not asymptotically flat (one would say it’s asymptoticallypathologic)

behaves like strong-field end-state of Wald’s charge accretion, see:W. A. Hiscock, On black holes in magnetic universes, J. Math. Phys.22, 1828-1833 (1981).

is complicated

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 6 / 9

Magnetised Kerr-Newman solution

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 7 / 9

Magnetised Kerr-Newman solution

However, the extremal case of MKN expels the magnetic field fromthe horizon (black hole Meissner effect)Thus, its horizon is just the Kerr-Newman oneNevertheless, there is backreaction. The near-horizon limit ischaracterised by effective parameters (J. Bičák, F.H.arXiv:1510.01911 [gr-qc], PhysRevD.92.104006):

M =√

Q2 + a2(

1 +14

B2Q2 + B2a2)+ BQa

a = a(

1− 34

B2Q2 − B2a2)− BQ

√Q2 + a2

Q = Q(

1− 14

B2Q2)+ 2Ba

√Q2 + a2

Quantities dϕ and A play a natural role in the near-horizon limitIt follows straightforwardly that the space of hyperbolas has the samestructure in the magnetised case

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 8 / 9

Acknowledgements

The work was supported by FCT (IDPASC) grant Bolsa deInvestigacao, reference PD/BD/113477/2015

Nao se esquecam de visitar os balneários pré-romanos antes deembarcar o comboio!

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 9 / 9

Acknowledgements/Thank you for your attention

The work was supported by FCT (IDPASC) grant Bolsa deInvestigacao, reference PD/BD/113477/2015Nao se esquecam de visitar os balneários pré-romanos antes deembarcar o comboio!

Filip Hejda (CENTRA-IST) The hyperbola revisited 27th May 2017, Braga 9 / 9