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Embeddings of the black holes in a flat space
Anton Sheykin, D. Grad and S.Paston
Saint-Petersburg State University

St. Petersburg 2013

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

1 / 19


Contents

What is an isometrical embedding Applications Exact embeddings of the black holes Thermodynamical properties of the black holes

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

2 / 19


Idea of embedding
Janet-Cartan theorem (1916)
n arbitrary n-dimensional Riemannian manifold can be locally isometrically embedded in N-dimensional flat space with N n(n + 1) . 2 (1)

For our 4D manifold N = 10; if the manifold has symmetries, N may be smaller. Metric of this manifold can be expressed in terms of embedding function: gµ = µ y a (x ) y b (x )ab , where y a (x ) ­ embedding function,
ab

(2)

­ metri of flat ambient space.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

3 / 19


Idea of embedding
Janet-Cartan theorem (1916)
n arbitrary n-dimensional Riemannian manifold can be locally isometrically embedded in N-dimensional flat space with N n(n + 1) . 2 (1)

For our 4D manifold N = 10; if the manifold has symmetries, N may be smaller. Metric of this manifold can be expressed in terms of embedding function: gµ = µ y a (x ) y b (x )ab , where y a (x ) ­ embedding function,
ab

(2)

­ metri of flat ambient space.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

3 / 19


Idea of embedding
Janet-Cartan theorem (1916)
n arbitrary n-dimensional Riemannian manifold can be locally isometrically embedded in N-dimensional flat space with N n(n + 1) . 2 (1)

For our 4D manifold N = 10; if the manifold has symmetries, N may be smaller. Metric of this manifold can be expressed in terms of embedding function: gµ = µ y a (x ) y b (x )ab , where y a (x ) ­ embedding function,
ab

(2)

­ metri of flat ambient space.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

3 / 19


Embedding-based theory of gravity
Change of variables in action: gµ (x µ ) y a (x µ ) d 4 x -g (G (3)

S =

1 2 =

µ

- T

µ

) gµ =

1 d 4 x -g (G µ - T µ )µ y a (x ) y b (x )ab = 2 = µ ( -g (G µ - T µ ) y a ) = (G µ - T µ )Dµ y a = 0. (4) (Regge, Teitelboim 1975) Natural appearance of the flat spacetime in this approach can be useful in the quantization of gravity: Preferred time slicing Definition of causality
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 4 / 19


Embedding-based theory of gravity
Change of variables in action: gµ (x µ ) y a (x µ ) d 4 x -g (G (3)

S =

1 2 =

µ

- T

µ

) gµ =

1 d 4 x -g (G µ - T µ )µ y a (x ) y b (x )ab = 2 = µ ( -g (G µ - T µ ) y a ) = (G µ - T µ )Dµ y a = 0. (4) (Regge, Teitelboim 1975) Natural appearance of the flat spacetime in this approach can be useful in the quantization of gravity: Preferred time slicing Definition of causality
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 4 / 19


Embedding-based theory of gravity
Change of variables in action: gµ (x µ ) y a (x µ ) d 4 x -g (G (3)

S =

1 2 =

µ

- T

µ

) gµ =

1 d 4 x -g (G µ - T µ )µ y a (x ) y b (x )ab = 2 = µ ( -g (G µ - T µ ) y a ) = (G µ - T µ )Dµ y a = 0. (4) (Regge, Teitelboim 1975) Natural appearance of the flat spacetime in this approach can be useful in the quantization of gravity: Preferred time slicing Definition of causality
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 4 / 19


Applications of isometrical embedding
Quantization
The Wheeler-de Witt equation for RT approach (Davidson, PRD 2003) 8 G h 8 G 2h
2

^ ( + R -
2

(3)

)(x )-
1 AB

^ ( - I )-

(x )

2 [y ] = 0 . (5) y (x ) y B (x )
A

Classification of exact solutions of Einstein equations
Embedding class p = N - d =
d (d -1) 2

of a metric is invariant.

Geometrical properties of Riemannian manifolds
Fronsdal embedding of the Schwarzchild black hole is closely related to the Kruskal coordinates:
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 5 / 19


Applications of isometrical embedding
Quantization
The Wheeler-de Witt equation for RT approach (Davidson, PRD 2003) 8 G h 8 G 2h
2

^ ( + R -
2

(3)

)(x )-
1 AB

^ ( - I )-

(x )

2 [y ] = 0 . (5) y (x ) y B (x )
A

Classification of exact solutions of Einstein equations
Embedding class p = N - d =
d (d -1) 2

of a metric is invariant.

Geometrical properties of Riemannian manifolds
Fronsdal embedding of the Schwarzchild black hole is closely related to the Kruskal coordinates:
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 5 / 19


Applications of isometrical embedding
Quantization
The Wheeler-de Witt equation for RT approach (Davidson, PRD 2003) 8 G h 8 G 2h
2

^ ( + R -
2

(3)

)(x )-
1 AB

^ ( - I )-

(x )

2 [y ] = 0 . (5) y (x ) y B (x )
A

Classification of exact solutions of Einstein equations
Embedding class p = N - d =
d (d -1) 2

of a metric is invariant.

Geometrical properties of Riemannian manifolds
Fronsdal embedding of the Schwarzchild black hole is closely related to the Kruskal coordinates:
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 5 / 19


Fronsdal embedding (1959)
r >R: y 0 = 2R y 1 = ±2R 1- R t sh , r 2R R t 1- ch , r 2R y y y y
2 3 4 5

r
= r cos(), = r sin() cos(), = r sin() sin(), = g (r ).

Ambient space metric is ab = diag(1, -1, -1, -1, -1, -1).
nton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 6 / 19


3 2.5 2

y

2

1.5 1 6 0.5 4 0 2 6 4 2 0 -2 -4 -6 -6 -4 -2 0

y

0

y1

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

7 / 19


Known embeddings of the Schwarzshild black hole
Kasner (1921)
y 0 = f (r ), y1 = R 1- R sin(t /R ), r y2 = R 1- R cos(t /R ). r (7)

Fujitani et al. (1961)
y0 = t 1- R , r y
1,2

1 = 2

2t 2

2

1

1-

R u (r ) + . r 2 (8)

Davidson and Paz (1999)
y
0,1

=

R 2 rc r

e

t + u (r )

r - rc e R

- t -u (r )

,

y 2 = kt .

(9)

y 3 = r cos ,

y 4 = r sin cos ,

y 5 = r sin sin .
St. Petersburg 2013 8 / 19

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )


New embeddings of the Schwarzshild black hole
Asymptotically flat embedding (S.Paston, A.S., CQG, 2012)
y 0 = t, y1 = y2 = 27R 3 sin r 27R 3 cos r t - 27R (r + 3R ) 27R 2 r
3

, .

(10)

t - 27R

(r + 3R )3 27R 2 r

Cubic embedding (S.P., A.S., TMPh, 2013)
y y y
0

1

2

2 3 R t + 1- t + u (r ), 6 2r R 2 = t 3 - t + u (r ), 6 2r 2 1 R = t+ 1- . 2 2 r =
St. Petersburg 2013

(11)

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

9 / 19


The global structure of the Schwarzchild black hole embeddings r=0
r =
IV

r

=

R

II

r

=

R

r

=



I

r

=



r

=

R
III r=0

r

=

R r =



I ­ our universe; II ­ black hole; III ­ white hole; IV ­ parallel universe. Fronsdal ­ I,II,III,IV; Kasner, Fujitani ­ I; Davidson, As. flat, Cubic ­ I,II.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

10 / 19


New global embeddings of the charged black hole
2m q 2 +2 r r dr 2 2m q 2 1- +2 r r

ds 2 =

1-

dt 2 -

- r 2 d 2 .

Spiral
y y y
0

= = =

(mr - q 2 )2 + b 2 r 2 sin (t + u (r )) , qr (mr - q 2 )2 + b 2 r 2 cos (t + u (r )) , qr b 2 + m2 - q 2 t. q
St. Petersburg 2013

1

2

(12)

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

11 / 19


Exponential
y y
0

1,2

= t, e - t -v = 2

(r )

1-

2m q 2 + 2 - r r

2

e

t +v ( r )

2

.

Cubic
y y
0

=-

q4 8m3

1- 1 4

1,2

= w(r ) -

2m q 2 2m 3 + 2 + 4 t 2, r r q 2 4m 6 2m q + 2 t+ t 1- r r 3q 8

3

t.

Ambient space metric is ab = diag(1, 1, -1, -1, -1, -1).
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 12 / 19


The global structure of the charged black hole embeddings

r=0

V

r

= r

r

-

r

VI = r-

r=0

II

r

=


IV

=

r+

r

=

r

+

r

=



I

r

=


III

r

=



Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

13 / 19


Hawking and Unruh effect
Hawking effect
A black hole has a radiation with a thermal spectrum T0 = 1 . 4 R

Tolman law
In thermal equilibrium a temperature is not constant when space is curved. For the Schwarzchild black hole T = T0 / 1 - R /r . (13)

Unruh effect
Uniformly accelerated observer, when coupled to the quantum fields, detects a radiation with a thermal spectrum and a temperature a proportional to his acceleration: T = . 2
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 14 / 19


Hawking and Unruh effect
Hawking effect
A black hole has a radiation with a thermal spectrum T0 = 1 . 4 R

Tolman law
In thermal equilibrium a temperature is not constant when space is curved. For the Schwarzchild black hole T = T0 / 1 - R /r . (13)

Unruh effect
Uniformly accelerated observer, when coupled to the quantum fields, detects a radiation with a thermal spectrum and a temperature a proportional to his acceleration: T = . 2
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 14 / 19


Hawking and Unruh effect
Hawking effect
A black hole has a radiation with a thermal spectrum T0 = 1 . 4 R

Tolman law
In thermal equilibrium a temperature is not constant when space is curved. For the Schwarzchild black hole T = T0 / 1 - R /r . (13)

Unruh effect
Uniformly accelerated observer, when coupled to the quantum fields, detects a radiation with a thermal spectrum and a temperature a proportional to his acceleration: T = . 2
Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University ) St. Petersburg 2013 14 / 19


Connection between Hawking and Unruh effect in ambient space
Deser & Levin, PRD 1999
Unruh radiation detected by observer moving on the embedding surface in the ambient space has the same spectrum and temperature as the Hawking aemb radiation from the horizon in the corresponding manifold: T = . This 2 hypothesis was tested for Fronsdal embedding.

Further development
Hong, GRG 2003 ­ charged black holes, Chen et al., JHEP 2004 ­ stationary motions, Santos et al., PRD 2004 ­ D -dimensional black holes,..

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

15 / 19


T = a/2 =

1 4 R 1 - R /r

,

T0 = T

1 - R /r = const .

(14)

3

y

2

2

1

a
4 2 4 2 0 -2 -2 -4 -6 -6 -4 0

6

0

6

y

0

y1

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

16 / 19


Accelerations for new embeddings
1 27Rr 1- 1- R r
2

Asymptotically flat: a = Davidson-Paz: a = 1- R r R 1- r

R +k r

Cubic a =

Corresponding temperatures (if we assume that the temperature and acceleration are always related as in the Unruh effect) violate the Tolman law and do not coincide with BH temperature!

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

17 / 19


For moving on the arbitrary trajectories the spectrum of Unruh radiation is not exactly thermal, but at sufficiently slowly varying accelerations (a/a2 << 1) the Unruh formula works good (Barbado & Visser, PRD, 2012). Exact spectrum was found to be non-thermal for the trajectories corresponding to lines of time in cubic (Letaw, PRD 1981) and Davidson-Paz (Abdolrahimi, arXiv:1304:4237) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

18 / 19


For moving on the arbitrary trajectories the spectrum of Unruh radiation is not exactly thermal, but at sufficiently slowly varying accelerations (a/a2 << 1) the Unruh formula works good (Barbado & Visser, PRD, 2012). Exact spectrum was found to be non-thermal for the trajectories corresponding to lines of time in cubic (Letaw, PRD 1981) and Davidson-Paz (Abdolrahimi, arXiv:1304:4237) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

18 / 19


Summary

Isometrical embedding is a powerful tool for studying of Riemannian manifolds. Some features of embeddings possibly can help to quantize gravity. Exact embeddings of black holes might be related to their thermodynamical properties. Mapping between Hawking effect and Unruh effect in ambient space holds only for hyperbolic (Fronsdal-like) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

19 / 19


Summary

Isometrical embedding is a powerful tool for studying of Riemannian manifolds. Some features of embeddings possibly can help to quantize gravity. Exact embeddings of black holes might be related to their thermodynamical properties. Mapping between Hawking effect and Unruh effect in ambient space holds only for hyperbolic (Fronsdal-like) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

19 / 19


Summary

Isometrical embedding is a powerful tool for studying of Riemannian manifolds. Some features of embeddings possibly can help to quantize gravity. Exact embeddings of black holes might be related to their thermodynamical properties. Mapping between Hawking effect and Unruh effect in ambient space holds only for hyperbolic (Fronsdal-like) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

19 / 19


Summary

Isometrical embedding is a powerful tool for studying of Riemannian manifolds. Some features of embeddings possibly can help to quantize gravity. Exact embeddings of black holes might be related to their thermodynamical properties. Mapping between Hawking effect and Unruh effect in ambient space holds only for hyperbolic (Fronsdal-like) embeddings.

Anton Sheykin, D. Grad and S.Paston (Saint-Petersburg the black holesin a flat space Emb eddings of State University )

St. Petersburg 2013

19 / 19