(Non-)Existence of Lorentz Frame Near Black Hole Singularity

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

T HEORETICAL L IMITS ON THE (N ON -)E XISTENCE OF L ORENTZ F RAME N EAR B LACK H OLE S INGULARITY ∗ V H Satheeshkumar Department of Physics Baylor University Waco, TX 76798-7316, ∗

based on my work with Douglas G Moore

Gulf Coast Gravity Meeting; Oxford, MS; April 19 & 20, 2013

V H S ATHEESHKUMAR T HEORETICAL L IMITS ON THE (N ON -)E XISTENCE OF L ORENTZ F RAME N EAR B LACK H OLE S INGULARITY

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

O UTLINE OF THE TALK

P RELUDE T IDAL F ORCES O UR R ESULTS S OME R EMARKS


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

G RAVITY AT H IGH E NERGIES

I

According to Einstein, gravity is strongest near black holes. This makes black holes the ultimate experimental testing grounds for General Relativity.

I

It seems very unlikely that the usual incremental increase of knowledge from a combination of theory and experiment will ever get us to the Planck scale.

I

Our best hope is an examination of fundamental principles, paradoxes and contradictions, and the study of gedanken experiments.


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

G RAVITY AT H IGH E NERGIES

I

General Relativity is known to break down at singularities.

I

From an order-of-magnitude argument, it is expected that quantum corrections become important when the curvature is of the order of Planck scale avoiding the singularity. Rαβ γδ R αβ γδ ≈

1 . 4 lPl

(1)

Before we show this rigourosly, let us do a quick review of the subject!


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

N EWTON DISCOVERS G RAVITY !


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

N EWTON DISCOVERS G RAVITY !


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

E QUIVALENCE P RINCIPLE Apple doesn’t fall far from the tree!


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

A N I NDIAN DISCOVERED GRAVITY BEFORE N EWTON !

... but did not live to write down his findings! V H S ATHEESHKUMAR T HEORETICAL L IMITS ON THE (N ON -)E XISTENCE OF L ORENTZ F RAME N EAR B LACK H OLE S INGULARITY

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

L ORENTZ I NVARIANCE

I

Lorentz invariance is built into the traditional Quantum Field Theories (QFT) by construction.

I

General Relativity (GR) has Lorentz invariance as a local symmetry through the Equivalence Principle.

I

The strong Equivalence Principle (SEP) states that in a freely falling, non-rotating laboratory occupying a small region of spacetime, the laws of physics are those of Special Relativity (SR).


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

G EODESIC D EVIATION In GR, the tidal force on body by a gravitating body is given by the equation for geodesic deviation. Such an equation that is valid in any coordinate system is given by D 2ξ a + R acbd ξ b x˙ c x˙ d = 0. Du a


(2)

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

T IDAL F ORCE The tidal force experienced by a freely falling observer is given by d 2 ζ αˆ ˆ = c 2 R αˆˆ0ˆ0δˆ ζ δ 2 dτ

(3)

where R αˆβˆγˆδˆ = R

µ e α )µ σ νρ (ˆ

eˆβ

σ

eˆγ

ν

(ˆ eδ )ρ

(4)

is the Riemann tensor in the tetrad frame. We consider all the four possible asymptotically flat Black Hole solutions in GR. I Schwarzschild [1916] I Reissner-Nordstr¨ om [1918] I Kerr [1963] I Kerr-Newman [1965] V H S ATHEESHKUMAR T HEORETICAL L IMITS ON THE (N ON -)E XISTENCE OF L ORENTZ F RAME N EAR B LACK H OLE S INGULARITY

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

S CHWARZSCHILD B LACK H OLE The Schwarzschild metric in the usual coordinates is given by 2µ 2µ −1 2 2 2 ds = 1 − c dt − 1 − dr − r 2 (dθ 2 + sinθ 2 dφ 2 ) r r 2

These lead to the following components of the mixed Riemann tensor in the tetrad basis. d 2 ζ ˆr dτ 2 ˆ d 2ζ θ dτ 2 ˆ d 2ζ φ dτ 2

2µ 2 ˆr c ζ r3 µ ˆ = − 3 c 2ζ θ r µ ˆ = − 3 c 2ζ φ r = +


(5)

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

¨ B LACK H OLE R EISSNER -N ORDSTR OM The Reissner-Nordstr¨om metric is given by −1 2µ q 2 2µ q 2 2 2 2 + 2 c dt − 1 − + 2 ds = 1− dr 2 r r r r

(6)

− r 2 (dθ 2 + sinθ 2 dφ 2 ) 2

GQ where q 2 = 4πε 4 These lead to the following components of the 0c mixed Riemann tensor in the tetrad basis. 2µ 3q 2 d 2 ζ ˆr = − c 2 ζ ˆr (7) dτ 2 r3 r4 ˆ d 2ζ θ µ q2 ˆ = − 3 − 4 c 2ζ θ 2 dτ r r ˆ 2 φ d ζ µ q2 ˆ = − 3 − 4 c 2ζ φ 2 dτ r r V H S ATHEESHKUMAR T HEORETICAL L IMITS ON THE (N ON -)E XISTENCE OF L ORENTZ F RAME N EAR B LACK H OLE S INGULARITY

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

K ERR B LACK H OLE The Kerr metric in the Boyer-Lindquist coordinates is given by ds 2 =

2 ρ 2 2 ∆ 2 − dr − ρ 2 dθ 2 c dt − a sinθ dφ ρ2 ∆ 2 2 2 2 2 (r + a ) sinθ a − dφ − 2 c dt ρ2 r + a2

(8)

where ρ 2 = r 2 + a2 cos 2 θ 2

∆ = r − 2µr + a

(9)

2

Σ2 = (r 2 + a2 )2 − a2 ∆ sin2 θ .


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

K ERR B LACK H OLE These lead to the following components of the mixed Riemann tensor in the tetrad basis. d 2 ζ ˆr 2µr 2 = (r − 3a2 cos2 θ ) (r 2 + a2 )2 + a4 sin4 θ (10) 2 10 dτ ρ −2a2 (r 2 + a2 ) sin2 θ c 2 ζ ˆr ˆ

d 2ζ θ dτ 2

= −

µr 2 (r − 3a2 cos2 θ ) (r 2 + a2 )2 + a4 sin4 θ 10 ρ ˆ −2a2 (r 2 + a2 ) sin2 θ c 2 ζ θ

ˆ

d 2ζ φ dτ 2

= −

µr 2 2 2 (r − 3a cos θ ) ∆(r 2 + a2 )2 + a4 ∆ sin4 θ ρ 10 ∆ ˆ −2a2 (r 2 + a2 )2 sin2 θ c 2 ζ φ


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

K ERR -N EWMAN B LACK H OLE In the Boyer-Lindquist coordinates, the Kerr-Newman metric is given by ds 2 =

2 ρ 2 2 ∆ 2 − dr − ρ 2 dθ 2 c dt − a sinθ dφ ρ2 ∆ 2 2 2 2 2 (r + a ) sinθ a − dφ − 2 c dt ρ2 r + a2

(11)

where ρ 2 = r 2 + a2 cos 2 θ 2

2

∆ = r − 2µr + a + q

(12) 2

Σ2 = (r 2 + a2 )2 − a2 ∆ sin2 θ . The symbols have their usual meanings, q 2 =

GQ 2 . 4πε0 c 4


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

K ERR -N EWMAN B LACK H OLE The tidal forces of a Kerr-Newman black hole are d 2 ζ ˆr dτ 2

=

ˆ

d 2ζ θ dτ 2

=

ˆ

d 2ζ φ dτ 2

=

i 1 h 2 2 2 2 2 2 2 2µr (r − 3a cos θ ) − 3q r − a cos θ (13) ρ 10 2 (r + a2 )2 + a4 sin4 θ − 2a2 (r 2 + a2 ) sin2 θ c 2 ζ ˆr i −1 h 2 2 2 2 2 2 2 − a cos θ − q r − 3a cos θ µr r ρ 10 2 ˆ (r + a2 )2 + a4 sin4 θ − 2a2 (r 2 + a2 ) sin2 θ c 2 ζ θ i −1 h 2 2 2 2 2 2 2 µr r − 3a cos θ − q r − a cos θ ρ 10 ∆ ˆ ∆(r 2 + a2 )2 + a4 ∆ sin4 θ − 2a2 (r 2 + a2 )2 sin2 θ c 2 ζ φ


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

O UR R ESULTS The complete gravitational collapse of a body always produces a Kerr-Newman black hole. Such a black hole is characterized by only three parameters namely its mass, electric charge and angular momentum. Let L be the length scale at which the tidal forces kick in and if l is the length scale of the theory, then for tidal forces not have an effect on the experiment, we should have l > rS lPl .


(19)

BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

QFT NEAR B LACK H OLE S INGULARITY Th freely-falling frame of size l will not feel the tidal forces or stays inertial only if it is at distance r from the singularity which is, 3 rmin >> rs l 2 .

(20)

The lower limits on distances (like Roche Limit) for a solar mass black hole for various length scales are summarized in the table. Expt. Type QED QCD EW Planck

∼l m 10−12 10−15 10−18 10−35

2.95 m 10−21 10−27 10−33 10−67

rmin >> rS 10−24 10−30 10−36 10−70

1.82 lp 1014 108 102 10−32


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

QFT NEAR B LACK H OLE H ORIZON We now concentrate on what happens at the Schwarzschild radius, rS , and derive limits on the mass of the black hole. Mmin >>

c2 l. 2G

(21)

The lower limits on the mass (like Chandrasekhar Limit) for various length scales are summarized in the table below. Expt. Type QED QCD EW Planck

∼l m 10−12 10−15 10−18 10−35

6.74 kg 1014 1011 108 10−9

Mmin >> 3.39 MSun 10−16 10−19 10−22 10−39

3.12 mp 1022 1019 1016 10−1


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

D EATH BY B LACK H OLE !

I

Assuming that the limit of tolerance to stretching or compression of a human body is an acceleration gradient of ≈ 400 ms −2 /m, for a human to survive the tidal forces at the Schwarzschild radius requires a very massive black hole with M ≥ 105 MSun

I

If you fell towards a supermassive black hole, with say M ≈ 109 MSun

I

If you fell towards a “small” black hole, of mass say 10MSun , you would be shredded by the tidal forces of the hole well before you reached the event horizon.


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

T HINK INSIDE THE BOX !


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

S OME R EMARKS I I

I

I

Our detailed calculation is in good agreement with the standard order-of-magnitude estimate. The presence of singularity will lead to contradiction of the Strong Equivalence Principle, i.e., the laws of physics will no longer reduce to that of SR very close to singularity. Hence we also cannot use the usual QFT in the Minkowski background. For any black hole of a solar mass of greater, one can consider the space in the vicinity of the horizon to be Rindler and the laws of Black Hole Thermodynamics hold. Theorem: If the singularities are present in a theory that is locally Lorentz invariant, then there will always be region around the singularity which cannot be explained by such a theory and hence will require a more fundamental theory to explain the physics inside of such a region.


BAYLOR U NIVERSITY

O UTLINE

P RELUDE

T IDAL F ORCES

O UR R ESULTS

S OME R EMARKS

T HANKS !

Thank you for your attention!


BAYLOR U NIVERSITY