Example: Vortex geometry

As an example we are now considering the vortex geometry which looks similar to a draining bathtub (see figure 12.3). The background velocity potential has the form

φ₀=−Aln(r/a)−B θ (12.31)

which leads to a velocity of the fluid flow of

v₀ⁱ = (∂^rφ, ∂^θφ) = (∂rφ, ∂_θφ/r²) =−

12.4. Example: Vortex geometry 73

Figure 12.2: Trapped surfaces formed by moving fluid. (Source: Barceló et al. - Analogue Gravity)

Figure 12.3: Vortex Geometry: The spirals are the streamlines of the fluid flow. The outer circle is the killing horizon of ∂_t^µ while the inner is the event horizon. The region in between is the ergo-region. (Source: Barceló et al. - Analogue Gravity)

and to

v₀²= A²+B²

r² (12.33)

Calculating the acoustic metric explicitly leads to ds²=− c²−A²+B²

r²

!

dt²−2A

r dr dt−2B dθ dt+dr²+r² dθ²+dz² (12.34) The killing horizon is at

rergo=

√A²+B²

c (12.35)

Since for the event horizon only the radial part counts we set B = 0 and get the event horizon

r_horizon= |A|

c (12.36)

Given this similarities with the Kerr black hole makes it an interesting and acceptable analogy.

75

A. Useful formulas

The following formulas were collected by Robert McNess and are available online at http://jacobi.luc.edu/Useful.html.

V

Contents

1 Curvature tensors 1

2 Conventions for Differential Forms 2

3 Euler Densities 3

4 Hypersurfaces 3

5 Sign Conventions for the Action 4

6 Hamiltonian Formulation 5

7 Conformal Transformations 5

8 Small Variations of the Metric 6

9 The ADM Decomposition 8

10 Converting to ADM Variables 10

1. Curvature tensors

Consider a d+ 1 dimensional manifold M with metric gµν. The covariant derivative on M that is metric-compatible with g_µν is∇µ.

Christoffel Symbols

Γ^λ_µν = 1

2g^λρ(∂_µg_ρν+∂_νg_µρ−∂_ρg_µν) (1) Riemann Tensor

R^λ_µσν =∂_σΓ^λ_µν−∂_νΓ^λ_µσ+ Γ^κ_µνΓ^λ_κσ−Γ^κ_µσΓ^λ_κν (2) Ricci Tensor

Rµν=δ^σ_λR^λµσν (3)

Schouten Tensor

S_µν = 1 d−1

R_µν− 1 2dg_µνR

(4)

∇^νS_µν =∇µS^ν_ν (5) Weyl Tensor

C^λµσν =R^λµσν+g^λνSµσ−g^λσSµν+gµσS^λν−gµνS^λσ (6)

1

Commutators of Covariant Derivatives

2. Conventions for Differential Forms

p-Form Components

LetM be a manifold with dimensiond+ 1 = 2nan even number. Normalized so thatχ(S²ⁿ) = 2.

Let Σ⊂ Mbe addimensional hypersurface whose embedding is described locally by an outward-pointing, unit normal vectorn^µ. Rather than keeping track of the signs associated withn^µ being either spacelike or timelike, we will just assume thatn^µ is spacelike. Indices are lowered and raised using g_µν andg^µν, and symmetrization of indices is implied when appropriate.

First Fundamental Form / Induced Metric on Σ

hµν=gµν−nµnν (27)

Projection onto Σ

⊥T^{µ ...}_{ν ...} =h^µ_λ. . . h^σ_ν. . . T^{λ ...}_{σ ...} (28) Second Fundamental Form / Extrinsic Curvature of Σ

K_µν =⊥(∇µn_ν) =h_µ^λh_ν^σ∇λn_σ= 1

2£_nh_µν (29)

3

Trace of Extrinsic Curvature

K=∇µn^µ (30)

‘Acceleration’ Vector

a^µ =n^ν∇νn^µ (31)

Surface-Forming Normal Vectors

n_µ = 1

pg^νλ∂_να∂_λα∂_µα ⇒ ⊥∇[µn_ν] = 0 (32)

Covariant Derivative on Σ compatible withh_µν

DµT^{α ...}_{β ...} =⊥ ∇µT^{α ...}_{β ...} ∀ T =⊥T (33) Intrinsic Curvature of (Σ, h)

[Dµ,Dν]A^λ =R^λσµνA^σ ∀ A^λ=⊥A^λ (34) Gauss-Codazzi

⊥Rλµσν = Rλµσν−KλσKµν+KµσKνλ (35)

⊥

R_λµσνn^λ

= DνKµσ− DσKµν (36)

⊥

R_λµσνn^λn^σ

= − LnK_µν+K_µ^λK_λν +Dµa_ν−a_µa_ν (37) Projections of the Ricci tensor

⊥(R_µν) =Rµν+Dµa_ν−a_µa_ν− LnK_µν−K K_µν+ 2K_µ^λK_{ν λ} (38)

⊥(Rµνn^µ) =D^µKµν− DνK (39) R_µνn^µn^ν = − LnK−K^µνK_µν+Dµa^µ−a_µa^µ (40) Decomposition of the Ricci scalar

R =R −K²−K^µνK_µν−2LnK+ 2Dµa^µ−2a_µa^µ (41) Lie Derivatives alongn^µ

£_nK_µν = n^λ∇λK_µν+K_λν∇µn^λ+K_µλ∇νn^λ (42)

⊥(£_nF^{µ ...}ν ...) = £_nF^{µ ...}ν ... ∀ ⊥F =F (43) 5. Sign Conventions for the Action

These conventions follow Weinberg, keeping in mind that he defines the Riemann tensor with a minus sign relative to our definition. They are appropriate when using signature (−,+, . . . ,+). The d+ 1-dimensional Newton’s constant is 2κ² = 16πG_d+1. The sign on the boundary term follows from our definition of the extrinsic curvature.

4

I_G= 1 Gauge Field Coupled to Particles

IM = − 1

Gravity Minimally Coupled to a Gauge Field I =

The canonical variables are the metric h_µν on Σ and its conjugate momenta π^µν. The momenta are defined with respect to evolution in thespacelike directionn^µ, so this is not the usual notion of the Hamiltonian as the generator of time translations. Momentum Conjugate tohµν

π^µν = ∂L_M

The dimension of spacetime is d+ 1. Indices are raised and lowered using the metricg_µν and its inverseg^µν. Metric

b

gµν=e^2σgµν (54)

5

Christoffel

bΓ^λ_µν= Γ^λ_µν+ Θ^λ_µν (55) Θ^λ_µν=δ^λ_µ∇νσ+δ^λ_ν∇µσ−gµν∇^λσ (56) Riemann Tensor

Rb^λ_µρν= R^λ_µρν+δ^λ_ν∇µ∇ρσ−δ^λ_ρ∇µ∇νσ+gµρ∇ν∇^λσ−gµν∇ρ∇^λσ (57) +δ^λ_ρ∇µσ∇νσ−δ^λ_ν∇µσ∇ρσ+g_µν∇ρσ∇^λσ−g_µρ∇νσ∇^λσ (58) +

gµρδ^λ_ν−gµνδ^λ_ρ

∇^ασ∇ασ (59)

Ricci Tensor

Rbµν = Rµν−gµν∇²σ−(d−1)∇µ∇νσ+ (d−1)∇µσ∇νσ (60)

−(d−1)g_µν∇^λσ∇λσ (61) Ricci Scalar

Rb=e^−2σ

R−2d∇²σ−d(d−1)∇^µσ∇µσ

(62) Schouten Tensor

Sb_µν = S_µν− ∇µ∇νσ+∇µσ∇νσ− 1

2g_µν∇^λσ∇λσ (63)

Weyl Tensor

Cb^λ_µρν =C^λ_µρν (64)

Normal Vector

b

n^µ=e^−σn^µ nb_µ =e^σn_µ (65) Extrinsic Curvature

Kb_µν=e^σ

K_µν+h_µνn^λ∇λσ

(66) Kb =e^−σ

K+d n^λ∇λσ

(67) 8. Small Variations of the Metric

Consider a small perturbation to the metric of the formg_µν →g_µν+δg_µν. All indices are raised and lowered using the unperturbed metric g_µν and its inverse. All quantities are expressed in terms of the perturbation to the metric with lower indices, and never in terms of the perturbation to the inverse metric. As in the previous sections, ∇µ is the covariant derivative on M compatible with g_µν and Dµ is the covariant derivative on a hypersurface Σ compatible withh_µν.

Inverse Metric

g^µν→g^µν−g^µαg^νβδg_αβ+g^µαg^νβg^λρδg_αλδg_βρ+. . . (68) 6

g→ g 1 +

9. The ADM Decomposition

The conventions and notation in this section (and the next) are different than what was used in the preceding sections. We consider a d-dimensional spacetime with metrich_ab.

We start by identifying a scalar fieldtwhose isosurfaces Σ_t are normal to the timelike unit vector given by

u_a=−α ∂_at , (84)

where the lapse function αis

α:= 1

p−h^ab∂at ∂bt . (85) An observer whose worldline is tangent tou_a experiences an acceleration given by the vector

a_b=u^c·^d∇cu_b , (86)

which is orthogonal to u_a. The (spatial) metric on thed−1 dimensional surface Σ_t is given by

σ_ab=h_ab+u_au_b . (87)

The intrinsic Ricci tensor built from this metric is denoted by Rab, and its Ricci scalar is R. The covariant derivative on Σ_t is defined in terms of thed dimensional covariant derivative as

D_aV_b :=σ_a^cσ_b^{e d}∇cV_e

for any V_b=σ_b^cV_c . (88)

The extrinsic curvature of Σ_t embedded in the ambientd dimensional spacetime (the constantr surfaces from the previous section) is

θ_ab:=−σacσ_b^{d d}∇cu_d

=−^d∇au_b−uaa_b =−1

2£uσ_ab . (89)

This definition has an additional minus sign, compared to the extrinsic curvatureK_µνfor the constantrsurfaces of the previous section. This is merely for compatibility with the standard conventions in the literature.

Now we consider a ‘time flow’ vector fieldt^a, which satisfies the condition

t^a∂_at= 1. (90)

The vectort^a can be decomposed into parts normal and along Σ_t as

t^a=α u^a+β^a , (91)

whereα is the lapse function (85) andβ^a :=σ^abt^b is the shift vector. An important result in the derivations that follow relates the Lie derivative of a scalar or spatial tensor (one that is orthogonal to u^a in all of its indices) along the time flow vector field, to Lie derivatives alongu^aandβ^a. LetS be a scalar. Then

£_tS=£_{α u}S+£_βS=α£_uS+£_βS . (92) Rearranging this expression then gives

£_uS= 1

α £_tS−£_βS

. (93)

Similarly, for a spatial tensor with all lower indices we have

£tWa...=α£uWa...+£_βWa... . (94) 8

vectorβ^a.

Next, we construct the coordinate system that we will use for the decomposition of the equations of motion.

The adapted coordinates (t, xⁱ) are defined by

∂_tx^a:=t^a. (95)

The xⁱ are ddimensional coordinates along the surface Σ_t. If we define P_i^a:= ∂x^a

∂xⁱ , (96)

then it follows from the definition of the coordinates thatP_i^a∂_at= 0 and we can useP_i^ato project tensors onto Σ_t. For example, in the adapted coordinates the spatial metric, extrinsic curvature, and acceleration and shift vectors are

σ_ij =P_i^aP_j^bσ_ab (97)

θij =PiaPjbθ_ab (98)

a_j =P_j^ba_b (99)

β_i =P_i^aβ_a=P_i^at_a. (100)

The line element in the adapted coordinates takes a familiar form:

habdx^adx^b= hab

Thus, in the adapted coordinate system we can express the components of the (d dimensional) metrich_ab and its inverseh^ab as

h_ab= −α²+βⁱβi σijβ^j

Obtaining the components of the inverse is a short algebraic calculation. Note that the spatial indices ‘i, j, . . .’

in the adapted coordinates are lowered and raised using the spatial metricσ_ij and its inverseσ^ij.

In adapted coordinates there are several results concerning the projections of Lie derivatives of scalars and tensors which will be important in what follows. The first, which is trivial, is that the Lie derivative of a scalar S along the time-flow vectort^ais just the regular time-derivative

£_tS=t^a∂_aS= ∂x^a

∂t

∂S

∂x^a =∂_tS . (109)

Next, we consider the projectorP_i^a applied to the Lie derivative alongt^a of a general vectorW_a, which gives

Pia£tWa=∂tWa ∀ Wa. (110)

9

The important point is that this applies not just to spatial vectors but to any vector W_a, as a consequence of the result

Pia£tua= 0. (111)

Finally, we can show that the Lie derivative alongt^a of any contravariant spatial vector satisfies

Pⁱ_a£_tV^a=∂_tVⁱ ∀ Vⁱ =Pⁱ_aV^a . (112) This follows from a lengthier calculation than what is required for the first two results.

Given these results, we can express various geometric quantities and their projections normal to and along Σ_t in terms of quantities intrinsic to Σ_t and simple time derivatives. First, the extrinsic curvature is

θ_ij = − 1

Sinceθab is a spatial tensor, projections of its Lie derivative alongu^a can be expressed in a similar manner P_i^aP_j^b£_uθ_ab= 1

α ∂_tθ_ab−£_βθ_ab

. (116)

Now we present the Gauss-Codazzi and related equations in adapted coordinates:

PiaPjb dRab

10. Converting to ADM Variables The metric is often presented in the form

h_abdx^adx^b =h_ttdt²+ 2h_tidtdxⁱ+h_ijdxⁱdx^j . (121) We would like to relate these components to the ADM variables: the lapse functionα, the shift vectorβi, and the spatial metricσij. This is a fairly straightforward exercise in linear algebra. Comparing with (105), we first note that

σij =hij . (122)

The inverse spatial metric,σ^ij, is literally the inverse ofh_ij, which isnot the same thing as h^ij

σ^ij = (σ_ij)⁻¹ = (h_ij)⁻¹ 6=h^ij . (123) For the shift vector we have

h_ti=σ_ijβ^j → σ^ikh_tk =σ^ikσ_klβ^l=βⁱ (124)

⇒βⁱ =σ^ijh_tj . (125)

Finally, for the lapse we obtain

α²=σ^ijh_tih_tj −h_tt . (126)

10

References

[1] C. M. Will, “The confrontation between general relativity and experiment,”Living Reviews in Relativity9 (2006), no. 3,.

[2] S. M. Carroll,Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, 2004.

[3] S. Hawking and G. Ellis,The Large scale structure of space-time. Cambridge University Press, 1973.

[4] P. K. Townsend, “Black holes: Lecture notes,”gr-qc/9707012.

[5] C. Barceló, S. Liberati, and M. Visser, “Analogue gravity,”Living Reviews in Relativity8 (2005), no. 12,.

Im Dokument Black Holes I VU (Seite 72-86)