De Felice F., Bini D. Classical Measurements in Curved Space-Times

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6.14 Physical properties of ﬂuids 123

In terms of the internal energy, the First Law of Thermodynamics becomes

Dˆ

dτ

= w

−

Θ(U).

One may also introduce the proper entropy s

per unit of proper mass,

dτ

= μ

, (6.224)

where T

is the temperature.

In the absence of external forces (f = 0 and hence w

= 0) one has the conser-

vation of the entropy density s

along the ﬂow lines of U,

dτ

=0,

and the acceleration of the ﬂuid lines reduces to

a(U)=−

+ p

∇(U)p

The relative point of view can be obtained directly from (6.205):

(fw,U,u)

= −

+ p

[P (U, u)∇p

− ν(ν ·P (U, u)∇p

)]

+ γ

−1

(G)

(fw,U,u)

− ν(ν ·F

(G)

(fw,U,u)

)]. (6.225)

Hydrodynamics with thermal ﬂux: absolute formulation

We can generalize our previous results to the case of a ﬂuid with thermal ﬂux.

We shall here illustrate only the general procedure, omitting unnecessary detail.

The energy-momentum tensor can be written as in (6.179) but having in addition

thermal stresses Q

≡ Q(U):

T = ρ

U ⊗ U + S

= T

%&'(

mechanical

+ Q

%&'(

thermal

where

= U ⊗ q

+ q

⊗ U,

with q

= q

(U)and

ALT S

=0, ALT X

=0,X

U =0,q

U =0.

124 Local measurements

Let us ﬁrst consider the splitting of the equations of motion div T = μ

f in the

comoving frame of the ﬂuid. Projection along U and onto LRS

gives

a(U)=μ

P (U)(f

+ f

)=μ

P (U)(f

(i)

+ f

(e)

+ f

(th)

dˆ

dτ

+(q

)

≡

, (6.226)

where (q

)

is the thermal conduction power in the comoving frame. We now

have a thermal force and a thermal power,

P (U)f

= −P (U )div Q, (6.227)

and

= −[μ

f − div S] · U =(μ

tot

− w

(i)

)=W

+ μ

)

, (6.228)

with

tot

= w

+(q

)

= U · div Q ≡−[∇·q

+ a(U) ·q

(i)

= −U · div T . (6.229)

Hydrodynamics with thermal ﬂux: relative formulation

Let us project the evolution equations of the proper frame along and orthogonally

to u. We ﬁnd

ˆμa

(fw,U,u)

= μP (u, U, u)

−1

tot

(fw,U,u)

− ν

, (6.230)

where

μF

tot

(fw,U,u)

= μ

P (u)[f

+ f

]. (6.231)

From the energy theorem we ﬁnd instead

dˆ

dτ

(U,u)

= γ

, (6.232)

where Q = Q

/γ.

Example: the viscous ﬂuid of Landau-Lifshitz

The viscous ﬂuid of Landau-Lifshitz (see Misner, Thorne, and Wheeler, 1973,

p. 567) is characterized by the following mechanical stress tensor:

=[p

− ζΘ(U)]P (U) − 2ησ(U), (6.233)

6.14 Physical properties of ﬂuids 125

where σ(U)=θ(U) −

Θ(U)P (U) is the trace-free part of the expansion ﬁeld.

We have

)

= −[div

+2a(U) · q

], (6.234)

(i)

= p

Θ(U) − [ζΘ(U)

+2ηTr(σ(U)

)]. (6.235)

Also in this case one introduces the scalar entropy s, related to the temperature

by the First Law of Thermodynamics (generalized to include the eﬀects of the

thermal action),

dτ

= q

tot

Θ(U) − w

(i)

≡ w

+(q

)

+[ζΘ(U)

+2ηTr(σ(U )

)], (6.236)

and the entropy 4-vector (see Exercise 22.7 of Misner, Thorne, and Wheeler,

1973)

S = μ

sU +

. (6.237)

Taking into account the mass conservation law, ∇

(μ

U) = 0, one ﬁnds

∇·S= μ

+[ζΘ(U)

+2ηTr(σ(U)

)] −q

· [∇ln T

+ a(U)]. (6.238)

In the absence of external forces we have f =0andw

= 0; therefore the above

expressions are considerably simpliﬁed.

Non-local measurements

The Principle of Equivalence states that gravitational and inertial accelerations

cannot be distinguished from each other if one neglects the curvature of the

background geometry. But even if curvature is taken into account, the choice of

the observer together with a frame adapted to him/her pollutes the measurements

with inertial contributions which are entangled with the curvature in a non-

separable way. The curvature is responsible for a relative acceleration among

freely falling particles. This acceleration induces a ﬁeld of strains which can be

measured with a suitable experimental device; however, a relative acceleration

also arises from orbital constraints if the bodies are not in free fall. In this case a

relativistically complete and correct description of the relative strains must also

take into account the properties of the observer and of his frame. A measurement

which carries the signature of the space-time curvature within its measurement

domain is termed non-local.

Here we shall ﬁrst identify the components of the space-time curvature relative

to a given frame and then discuss various ways to measure them.

7.1 Measurement of the space-time curvature

Let u be a vector ﬁeld whose integral curves form a congruence C

which we

assume to be representative of a family of observers. With respect to the latter,

then, the spatial and temporal splitting of the Riemann tensor identiﬁes the

following three spatial ﬁelds:

E(u)

αβ

= R

αμβν

H(u)

αβ

= −R

∗

αμβν

F(u)

αβ

∗

]

αμβν

, (7.1)

where E(u)

[αβ]

=0=F(u)

[αβ]

and H (u)

= 0. The ﬁrst two quantities are

termed, respectively, the electric and magnetic parts of the Riemann tensor. The

7.1 Measurement of the space-time curvature 127

20 independent components of the Riemann tensor are then summarized by the

6 independent components of the electric part (spatial and symmetric tensor), the

8 independent components of the magnetic part (spatial and trace-free tensor),

and the 6 independent components of the mixed part (spatial and symmetric

tensor).

Consider a frame {e

} (not necessarily orthonormal) adapted to the observer

u so that u = e

with e

a basis in LRS

. Then all the above spatial quantities

can be written as

E(u)

= R

a0b0

H(u)

= −R

∗

a0b0

η(u)

a0cd

F(u)

∗

]

a0b0

η(u)

cdef

, (7.2)

and can be inverted to give

= H(u)

η(u)

bcd

abcd

= η(u)

abr

η(u)

cds

F(u)

. (7.3)

Using these relations one has also the frame components of the Ricci tensor

= R

μα

μβ

= −E(u)

= η(u)

abc

H(u)

= −E(u)

−F(u)

+ δ

F(u)

, (7.4)

so that

R = R

+ R

= −2(E(u)

−F(u)

). (7.5)

Asshownin(2.66), the Riemann tensor can be written in terms of the Weyl

tensor, the Ricci tensor, and the scalar curvature; in four dimensions we have

αβ

γδ

= C

αβ

γδ



[γ

δ]

− δ

[γ

δ]



= C

αβ

γδ

+2δ

[α

[γ

δ]

β]

, (7.6)

where

αβ

= R

αβ

−

αβ

. (7.7)

Clearly, in vacuum (R

αβ

=0,R = 0), the Weyl and Riemann tensors coincide.

Similarly to the Riemann tensor, the splitting of the Weyl tensor identiﬁes the

following two spatial ﬁelds, because of the identity

∗

= −C (or

∗

C = C

∗

E(u)

αβ

= C

αμβν

,H(u)

αβ

= −C

∗

αμβν

. (7.8)

Tensors E(u)andH(u) are termed, respectively, the electric and magnetic parts

of the Weyl tensor and are related to E(u), H(u), and F(u) by the following

relations:

128 Non-local measurements

E(u)=

[E(u) −F(u)]

(TF)

H(u) = SYM H(u), (7.9)

from (7.6). Written in terms of components with respect to a frame adapted to

u,wehave

E(u)



E(u)

−F(u)

−

(E(u)

−F(u)

)



H(u)

= H(u)

(ab)

. (7.10)

In Chapter 3 we evaluated the components of the Riemann tensor in an adapted

frame; in fact expressions (3.92), (3.97), and (3.98) are equivalent to

E(u)

=[∇(u)

+ a(u)

]a(u)

+ ∇(u)

(fw)

k(u)

− [k(u)

]

H(u)

= −



∇(u)

k(u)

η(u)

bcd

+2a(u)

ω(u)



F(u)

=[θ(u)

− Θ(u)θ(u)]

−

[Trθ(u)

− Θ(u)

]

+3ω(u)

ω(u)

− G

(sym)

, (7.11)

where

(sym)

= R

(sym)

−

(sym)

, (7.12)

with R

(sym)

= R

(sym)

and R

(sym)

= R

(sym)

from (3.105). Furthermore,

E(u)

=[∇(u)

+ a(u)

]a(u)

−∇(u)

(fw)

Θ(u)+2ω(u)

ω(u)

− Tr θ(u)

H(u)

=2[∇(u)

− a(u)

]ω(u)

≡ 0,

F(u)

= −

[Tr θ(u)

− Θ(u)

]+3ω(u)

ω(u)

(sym)

, (7.13)

where the trace of H(u) vanishes identically from (3.82). From (7.11) and recalling

the deﬁnition of the symmetric curl (see Eqs. (3.40) and (3.41)), we have

H(u)

= H(u)

(ab)

= −[Scurl

k(u)]

− 2a(u)

ω(u)

= −[Scurl

ω(u)]

+ [Scurl

θ(u)]

− 2a(u)

ω(u)

[Scurl

θ(u)]

− 2a(u)

ω(u)

−∇(u)

ω(u)

(7.14)

and

E(u)

[∇(u)

a(u)

+ a(u)

a(u)

−∇(u)

(fw)

θ(u)

+2ω(u)

ω(u)

− 2[θ(u)

]

+Θ(u)θ(u)

− R

(sym)ab

]

(7.15)

7.2 Vacuum Einstein’s equations in 1+3 form 129

If u is expansion-free, the electric and magnetic parts of the Weyl tensor

reduce to

E(u)

[∇(u)

a(u)

+ a(u)

a(u)

+2ω(u)

ω(u)

− R

(sym)ab

]

H(u)

= −[2a(u)

ω(u)

+ ∇(u)

ω(u)

]

(7.16)

7.2 Vacuum Einstein’s equations in 1+3 form

Einstein’s equations without the cosmological constant are given by

αβ

−

αβ

=8πT

αβ

. (7.17)

In the absence of matter energy sources (T

αβ

= 0) they become R

αβ

=0.Using

(7.4) we can write these equations in terms of the kinematical parameters of the

observer congruence C

and their derivatives. From (7.2)

and with respect to a

frame adapted to u,wehave

E(u)

=0 (R

=0),

H(u)

[ab]

=0 (R

=0),

E(u)

+ F(u)

= δ

F(u)

=0).

(7.18)

Using the trace-free property of E(u) in the last equation we ﬁnd F(u)

=0so

that from the above equations we deduce that

E(u)

= −F(u)

. (7.19)

Moreover, from (7.13)

, the condition R

= 0 gives

[∇(u)

+ a(u)

]a(u)

−∇(u)

(fw)

Θ(u)+2ω(u)

ω(u)

− Trθ(u)

=0. (7.20)

From (7.4)

, the components R

= 0 can be written in the form

cab

H(u)

=0, (7.21)

which is equivalent to

−∇(u)

[θ(u)

− Θ(u)δ

] −[curl

ω(u)]

− 2[a(u) ×

ω(u)]

=0. (7.22)

Finally, R

= 0 gives

− [∇(u)

(lie)

+Θ(u)]θ(u)

− [∇(u)

+ a(u)

]a(u)

+ R

(sym)

−2[ω(u)

ω(u)

− δ

ω(u)

]=0. (7.23)

130 Non-local measurements

This set of equations takes a simpliﬁed form when the observer congruence is

Born-rigid, i.e. θ(u) = 0. In this case we have

[∇(u)

+ a(u)

]a(u)

+2ω(u)

ω(u)

=0, (7.24)

[curl

ω(u)]

+2[a(u) ×

ω(u)]

=0, (7.25)

(sym)

− [∇(u)

+ a(u)

]a(u)

− 2[ω(u)

ω(u)

−P(u)

ω(u)

]=0. (7.26)

Contracting the indices in (7.26) yields

(sym)

−∇(u)

a(u)

− a(u)

a(u)

+4ω(u)

ω(u)

= 0 (7.27)

which, using (7.24), leads to

(sym)

+6ω(u)

ω(u)

=0. (7.28)

7.3 Divergence of the Weyl tensor in 1+3 form

The divergence of the Weyl tensor ∇

αβγ

in an adapted frame (1 + 3 form) is

represented by the following independent ﬁelds:

∇

0a0

, ∇

(ab)0

, ∇

∗

0a0

, ∇

∗

(ab)0

. (7.29)

Other components give non-independent relations due to the trace-free property

of the Weyl tensor and the Ricci identity. It is suﬃcient to deduce the 1 + 3 form

of the ﬁrst pair of the above relations because the remaining two are obtained by

the substitution E(u) → H(u)andH(u) →−E(u), similar to what is done for

the electromagnetic ﬁeld. We have

∇

0a0

= e



0a0



+Γ

μδ

0a0

− Γ

0δ

σa0

−Γ

aδ

0σ0

− Γ

0δ

0aσ

. (7.30)

Substituting the values (3.71) of the connection coeﬃcients of an adapted frame

and using deﬁnitions (7.8), this equation can be cast in the form

∇

0a0

div

E(u) −2

∗(u)

[θ(u) H(u)] −2H(u) ω(u)

. (7.31)

Similarly we have

∇

(ab)0

= ∇(u)

(fw)

E(u)

− [Scurl

H(u)]

+2Θ(u)E(u)

−2[a(u) ×

H(u)]

− [ω(u) ×

E(u)]

−3[SYM(θ(u) E(u))]

. (7.32)

Summarizing, the set of equations representing the divergence of the Weyl tensor

is the following

∇

0a0

= {div

E(u) −2

∗

(u)

[θ(u) H(u)] −2H(u) ω(u)}

∇

∗

0a0

= {div

H(u)+2

∗

(u)

[θ(u) E(u)] + 2E(u) ω(u)}

7.4 Electric and magnetic parts of the Weyl tensor 131

∇

(ab)0

= ∇(u)

(fw)

E(u)

− [Scurl

H(u)]

+2Θ(u)E(u)

−2[a(u) ×

H(u)]

− [ω(u) ×

E(u)]

−3[SYM(θ(u) E(u))]

∇

∗

(ab)0

= ∇(u)

(fw)

H(u)

+ [Scurl

E(u)]

+2Θ(u)H(u)

+2[a(u) ×

E(u)]

− [ω(u) ×

H(u)]

−3[SYM(θ(u) H(u))]

. (7.33)

In vacuum (T

= 0) and with respect to an observer u described by an expansion-

free (θ(u) = 0) congruence of integral curves, and from (7.8), we can write

Einstein’s equations in Maxwell-like form (compare with (6.159)–(6.162)):

0 = div

E(u) −2H(u) ω(u),

0 = div

H(u)+2E(u) ω(u),

0=∇(u)

(fw)

E(u) −[Scurl

H(u)] − 2[a(u) ×

H(u)] − [ω(u) ×

E(u)],

0=∇(u)

(fw)

H(u) + [Scurl

E(u)] + 2[a(u) ×

E(u)] −[ω(u) ×

H(u)]. (7.34)

7.4 Electric and magnetic parts of the Weyl tensor

Let U and u be two diﬀerent families of observers related by a boost

U = γ(U, u)[u + ν(U, u)]. (7.35)

Both of these observers can be used to split the Weyl tensor (and similarly the

Riemann tensor) in terms of associated electric and magnetic parts,

E(u)

αβ

= C

αμβν

,H(u)

αβ

= −C

∗

αμβν

E(U)

αβ

= C

αμβν

,H(U)

αβ

= −C

∗

αμβν

. (7.36)

Using (7.35) we have, for example,

E(U)

αβ

= γ(U, u)

αμβν

+ ν(U, u)

][u

+ ν(U, u)

], (7.37)

that is, abbreviating γ(U, u)=γ and ν(U, u)=ν,

E(U)

αβ

= γ

[E(u)

αβ

+ C

αμβν

+ C

αμβν

+ C

αμβν

]. (7.38)

Let e

denote a frame adapted to u (namely u = e

and e

with a =1, 2, 3

spanning the local rest space of u). We then have

E(U)

= γ

[E(u)

+ C

a0bc

+ C

acb0

+ C

acbd

]

= γ

[E(u)

+ H(u)

η(u)

+ H(u)

η(u)

−E(u)

η(u)

E(U)

= γ

0b0c

]=γ

[E(u)

E(U)

= γ

0cb0

+ C

0cbd

]

= γ

[−E(u)

− H(u)

η(u)

]. (7.39)

132 Non-local measurements

Therefore, in compact form, we have

[P (u, U)E(U )]

αβ

= γ

[E(u)

αβ

+2H(u)

(α|f|

η(u)

β)c

−E(u)

η(u)

αc

η(u)

βd

] (7.40)

and similarly

[P (u, U)H(U)]

αβ

= γ

[H(u)

αβ

− 2E(u)

(α|f|

η(u)

β)c

−H(u)

η(u)

αc

η(u)

βd

]. (7.41)

7.5 The Bel-Robinson tensor

In the theory of general relativity, gravitation is described as the curvature of the

background geometry; hence any manifestation of pure gravity is presented in

terms of the Riemann tensor. Neglecting the contribution to gravity by the local

distribution of matter-energy (that is, setting R

αβ

= 0), a possible generalization

of the energy-momentum tensor to the gravitational ﬁeld is the Bel-Robinson

tensor (Bel, 1958), deﬁned by

αβ

γδ

αρβσ

γρδσ

∗

αρβσ

∗

γρδσ

). (7.42)

In standard terminology, we use super-energy density and super-Poynting vector

(Maartens and Bassett, 1998) in reference to the analogous contractions in the

electromagnetic case relative to a general observer u, namely

(g)

(u)=T

αβγδ

[E(u)

+ H(u)

(g)

(u)

= T

αβγδ

=[E(u) ×

H(u)]

, (7.43)

where the notation (3.39) has been used. Similarly to what was done for the elec-

tromagnetic ﬁeld in Sections 6.11 and 6.12 with the electric and magnetic parts

of the Weyl tensor, we can introduce the complex (symmetric trace-free) spatial

tensor ﬁeld Z

(g)

(u)=E(u) − iH(u), abbreviated to Z(u) in this section, and

evaluate the eﬀect of a boost in a general direction U, diﬀerent from u,interms

of the orthogonal decompositions with respect to the relative spatial velocity of

the observers u and U . Z(u) can be decomposed into a scalar Z



(u), a vector

⊥

(u), and a tensor Z

⊥⊥

(u), the latter two being orthogonal to ˆν(U, u), i.e.

Z(u)=Z



(u)ˆν(U, u) ⊗ ˆν(U, u)+Z

⊥

(u) ⊗ ˆν(U, u)

+ˆν(U, u) ⊗ Z

⊥

(u)+Z

⊥⊥

(u), (7.44)

where Z

⊥⊥

(u) can then be further decomposed into its pure-trace part involving

Tr Z

⊥⊥

(u)=−Z



(u) and its trace-free part Z

⊥⊥(TF)

(u).