Finslerian extension of the Schwarzschild metric

Z.K. Silagadze

http://arxiv.org/abs/1007.4632

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

General Very Special Relativity is Finsler Geometry

G. W. Gibbons, J. Gomis and C. N. Pope, Phys. Rev. D 76, 081701(2007), arXiv:0707.2174.

Bogoslovsky metric ds2 = (nσdxσ)2b(ηµνdxµdxν)1−b.

What is a curved-space generalization of this metric?

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Radial metric in the Kruskal-Szekeres coordinates

Schwarzschild metric:

ds2 = α dT 2 − α−1 dR2 − R2(dθ2 + sin2 θ dφ2),

Where (units are such that c = 1 and G = 1)

α = 1− 2m

R.

Kruskal-Szekeres coordinates t, x (in the exterior region R > 2m):

t

x= tanh

(T

4m

),

(R

2m− 1

)eR/2m = x2 − t2.

Radial Schwarzschild metric in the Kruskal-Szekeres coordinates:

ds2 =32m3

Re−R/2m (dt2 − dx2).

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finslerian generalization of the radial metric

ds2 =32m3

Re−R/2m

(dt − dx

dt + dx

)b

(dt2 − dx2),

where R(x , t) is implicitly determined through the relation(R

2m− 1

)eR/2m =

(x − t

x + t

)b

(x2 − t2).

Invariant under the Bogoslovsky transformations

x ′ = e−bψ (x coshψ − t sinhψ),

t ′ = e−bψ (t coshψ − x sinhψ).

R ′ = R, T ′ = T − 4mψ.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finslerian radial metric in Schwarzschild coordinates

auxiliary variables u = x + t and v = x − t.

ds2 = 16m2α

(du

u

)1−b (−dv

v

)1+b

.

(R2m − 1

)eR/2m = u1−b v1+b −→ α−1 dR

2m = (1− b) duu + (1 + b) dv

v

u−vu+v = t

x = tanh(

T4m

)−→ dT

2m = duu −

dvv

du

u=

1

2

[(1 + b)

dT

2m+ α−1 dR

2m

],

−dv

v=

1

2

[(1− b)

dT

2m− α−1 dR

2m

].

ds2 = [(1−b)α1/2 dT−α−1/2 dR]2b [(1−b2)α dT 2−α−1 dR2−2b dR dT ]1−b.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

generalized tortoise coordinate

R∗ = R + 2m ln

(R

2m− 1

)+ bT

dR∗ = α−1 dR + b dT

ds2 =

(dT − dR∗dT + dR∗

)b

α (dT 2 − dR2∗ )

assympotic limit when R →∞:

ds2 →(

dT − dR∗dT + dR∗

)b

(dT 2 − dR2∗ ) =

(dT − dr

dT + dr

)b

(dT 2 − dr2).

r = R + bT

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Fronsdal embedding

z1 = 4m√α sinh

(T

4m

), z2 = 4m

√α cosh

(T

4m

), z3 = g(R),

z4 = R sin θ cosφ, z5 = R sin θ sinφ, z6 = R cos θ.

(dg

dR

)2

=2m

R+

(2m

R

)2

+

(2m

R

)3

= α−1

[1−

(2m

R

)4]− 1.

z22 − z2

1 = (4m)2(1− 2m

R

), z3 = g(R), z2

4 + z25 + z2

6 = R2

ds2 = dz21 − dz2

2 − dz23 − dz2

4 − dz25 − dz2

6

t =1

4m

√R

2mexp

(R

4m

)z1, x =

1

4m

√R

2mexp

(R

4m

)z2.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Fronsdal embedding – θ, φ = const surface

C. Fronsdal, Completion and Embedding of the Schwarzschild Solution,Phys. Rev. 116 (1959), 778-781

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finslerian generalization of the Fronsdal embedding

z0 = bT , z1 = 4m√

1− b2√α sinh

(T

4m

), z3 = f (R),

z2 = 4m√

1− b2√α cosh

(T

4m

), z4 = (R + bT ) sin θ cosφ,

z5 = (R + bT ) sin θ sinφ, z6 = (R + bT ) cos θ.

(df

dR

)2

= α−1

[1− (1− b2)

(2m

R

)4]− 1.

t = 14m

√R2m exp

(R+bT

4m

)z1√1−b2

, x = 14m

√R2m exp

(R+bT

4m

)z2√1−b2

.

(R

2m− 1

)eR/2m =

(x − t

x + t

)b

(x2 − t2),

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finslerian metric in the ambient space

ds2 = (NA dzA)2b (ηAB dzAdzB)1−b

ηAB = diag(+1,+1,−1,−1,−1,−1,−1)

NA NA = 0 – NA determines the null-direction

Metric axially symmetric −→ N4 = N5 = 0

NA(R,T ) such that in the case θ = π/2, φ = const the (Finslerian) radialmetric is induced.

b(N0 − N6) + α1/2√

1− b2

(N1 cosh

T

4M− N2 sinh

T

4M

)= (1− b)α1/2,(

2m

R

)2√

1− b2

α

(N1 sinh

T

4M− N2 cosh

T

4M

)− N3

df

dR− N6 = −α−1/2,

N20 + N2

1 − N22 − N2

3 − N26 = 0,

b = 1 −→ N0 = N6, N3dfdR + N6 = α−1/2.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

The “unique” solution

N0 = N6, N1 =

√1− b

1 + bcosh

T

4m, N2 =

√1− b

1 + bsinh

T

4m, N3 =

√1− b

1 + b,

N6 = α−1/2 − N3df

dR= α−1/2

1−√

1− b

1 + b

√2m

R

√1− (1− b2)

(2m

R

)3 .

ds2 = (NA dzA)2b (ηAB dzAdzB)1−b

ηAB dzAdzB = (1−b2)α dT 2−α−1 dR2−2b dR dT−(R+bT )2 (dθ2+sin2 θ dφ2),

NA dzA = (1− b)α 1/2 dT − α−1/2 dR + N6 d [(R + bT )(1− cos θ)].

In the asymptotic limit R →∞, this space-time has the Bogoslovsky metric:

ηAB dzAdzB → dT 2 − dr2 − r2(dθ2 + sin2 θ dφ2),

NA dzA → dT − d(r cos θ).

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finsler geometry

The fundamental idea goes back to Riemann, 1854.

ds = F (x1, · · · , xn, dx1, · · · , dxn)

Riemann geometry F 2 = gij(x)dx i dx j

Finsler geometry: every curve x(t) has a length derived from aninfinitesimal length or line element L =

∫F (x , x) dt.

F (x , dx) is smooth,F (x , dx) ≥ 0 with equality if and only if dx = 0 (positive definiteness),F (x , λdx) = λF (x , dx) for all λ ≥ 0 (homogeneity),F (x , dx + dy) ≤ F (x , dx) + F (x , dy) for all dy at the same tangentspace with dx .

Berwald-Moore metric. F 4 = dx1 dx2 dx3 dx4.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finsler, Caratheodory, Riemann, Cartan

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Riemann on Finsler geometry

“The next simplest case would comprise the manifolds, in which the lineelement can be expressed as the fourth root of a biquadratic differentialform. The investigation of these more general types would not require anyessentially different principles, but it would be time consuming andcontribute comparatively little new to the theory of space, because theresults cannot be interpreted geometrically” - Riemann, 1854.

“Here is one of the few instances where Riemann’s feeling was wrong” -Herbert Busemann, 1948.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Busemann on Finsler geometry

The term "Finsler space"evokes in most mathematicians the picture of animpenetrable forest whose entire vegetation consists of tensors.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finsler geometry - not so exotic

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Finsler geometry in the market

A. Kristaly, Gh. Morosanu and A. Roth, Optimal placement of a depositbetween markets: Riemann-Finsler geometrical approach, J. Optim. TheoryAppl. 139 (2008), 263-276.

cars that transport products from deposit to markets move alongstraight roads;the Earth gravity acts on them L = vt + g

2 sinα cos θ t2;the transport costs is proportional to the time elapsed to arrive fromdeposit to markets.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Matsumoto metric

M. Matsumoto, A slope of a mountain is a Finsler surface with respect to atime measure, J. Math. Kyoto Univ. 29N1 (1989) 17-25.

u = v + a cos θ, a = kg sinα.

t = Lu , L =

√(∆xa)2 + (∆ya)2, cos θ = ∆xa

L .

F (dx , dy) =dx2 + dy2

v√

dx2 + dy2 + a dx.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Dog-and-rabbit chase

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Dog-and-rabbit chase - duration of the chase

Z.K.S, G.I. Tarantsev, Eur. J. Phys. 31 (2010), L37-L38, arXiv:0909.2324.

~r = ~r2 −~r1 is parallel to the dog’s velocity.Hence it is perpendicular to the dog’s acceleration ~V1.

d

dt

[~r · (~V1 + ~V2)

]= (~V2 − ~V1) · (~V1 + ~V2) = V 2

2 − V 21 .

T =

[~r · (~V1 + ~V2)

]∣∣∣t=0

V 21 − V 2

2

=LV1

V 21 − V 2

2

.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Dog-and-rabbit chase - in the rabbit’s frame

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Zermelo navigation problem

Find minimum time trajectories in a Riemannian manifold with metric hij

under the influence of a drift (wind) represented by a vector field W i .

For time independent wind, the trajectories which minimize travel time areexactly the geodesics of a particular Finsler geometry, known as Randersmetric (Z. Shen, 2001, arXiv:math/0109060).

F (x , x) =√

aij(x)x i x j + bi (x)x i .

aij =λhij + WiWj

λ2, λ = 1−WiW

i , Wi = hijWj , bi = −Wi

λ.

G. W. Gibbons et al., Stationary Metrics and OpticalZermelo-Randers-Finsler Geometry, Phys. Rev. D79 (2009), 044022,arXiv:0811.2877.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Magnetic field and the Randers metric

The Lagrangian of relativistic electrons in a magnetic field gives rise to aRanders metric (R. Ingarden, 1957)

bi =q√2mε

Ai , ε =m

2aij

dx i

dτ

dx j

dτ.

Magnetic flow B i in a Riemanian metric aij

↑↓

Finslerian flow in a Randers metric aij , bi

↑↓

Zermalo navigation problem hij ,Wi

G. W. Gibbons et al., Phys. Rev. D79 (2009), 044022, arXiv:0811.2877.

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Schwarzschild metric - formal derivation

dl2 = dr2 + r2(dθ2 + sin2 θ dφ2) - Euclidean metric in polar coordinates.

ds2 = eA(r)dt2 − eB(r)dr2 − r2(dθ2 + sin2 θ dφ2).

Rµν = 0 - Einstein equations in vacuum.

ds2 =

(1− 2m

r

)dt2 −

(1− 2m

r

)−1

dr2 − r2(dθ2 + sin2 θ dφ2).

Units such that c=1,G=1.

Schwarzschild metric – the unique vacuum solution with sphericalsymmetry (Birkhoff’s theorem).

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Black holes - a waterfall analogy

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

The Schwarzschild waterfall

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Heuristic approach to the Schwarzschild geometry

M. Visser, Int. J. Mod. Phys. D14 (2005), 2051-2068, arXiv:gr-qc/0309072

ds2 = dt2in − d~r2

in, dtin = dt, d~rin = d~R − ~vt, V =√

2mR

ds2 =

(1− 2m

R

)dt2 − 2

√2m

RdR dt + d~R2

Schwarzschild metric in the Painleve-Gullstrand coordinates!

dT = dt −√

2m

Rα−1dR

ds2 = α dT 2 − α−1 dR2 − R2(dθ2 + sin2 θ dφ2)

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

Acoustics in fluids and general relativity– very deep and powerful analogies

W. G. Unruh, Experimental black hole evaporation? Phys. Rev. Lett, 46(1981), 1351–1353.M. Visser, Acoustic propagation in fluids: an unexpected example ofLorentzian geometry, gr-qc/931102.

propagation of sound waves ∂2t ψ = c2∇2ψ

if the fluid is non-homogeneous and in motion:

∆ψ ≡ 1√−g

∂µ(√−ggµν∂νψ

)= 0

g00 = ρc (c2 − v2), g0i = gi0 = v i , gij = −δij

The fluid particles couple only to the physical flat metric.Sound waves do not “see” the physical metric at all.

They couple only to the acoustic metric gµν .

Z.K. Silagadze Finslerian extension of the Schwarzschild metric

И Шахразаду застигло утро, и она прекратиладозволенные речи.

Here Shahrazad perceived the light of morning,and discontinued the recitation with whichshe had been allowed thus far to proceed.

�� ��The End

Z.K. Silagadze Finslerian extension of the Schwarzschild metric