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

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6.11 Local properties of an electromagnetic ﬁeld 113

Recalling that the physical meaning of a 4-vector is encoded in its magnitude,

we can deduce that the magnitude of P (u)f(U) is just the magnitude of the

Lorentz force, and so the moduli of the electric and the magnetic parts describe

the electric ﬁeld intensity and the magnetic induction.

By introducing the complex vector ﬁeld Z(u)=E(u) − iB(u) (Landau and

Lifshitz, 1975), Eqs. (6.142) and (6.143) can be written together as

(lrs)

(u, U) Z(U)]



= Z



(u),

(lrs)

(u, U) Z(U)]

⊥

=coshαZ

⊥

(u)

+ i sinh α [ˆν(U, u) ×

⊥

(u)], (6.144)

where ||ν(U, u)|| = tanh α. In terms of components with respect to an observer-

adapted frame {e

} (e

= u, with e

along ˆν(U, u)), this complex vector repre-

sentation reﬂects the isomorphism between the group of proper orthochronous

Lorentz transformations and SO(3,C), the group of proper orthogonal transfor-

mations. In the 2-plane orthogonal to e

within LRS

one has



(lrs)

(u, U) Z(U)]

⊥

(lrs)

(u, U) Z(U)]

⊥





cosh α −i sinh α

i sinh α cosh α



⊥

(u)

⊥

(u)



. (6.145)

6.11 Local properties of an electromagnetic ﬁeld

An interesting application is the splitting of the energy-momentum tensor of an

electromagnetic ﬁeld. We have

αβ

4π



αρ

−

αβ

ρσ



4π

]

αβ

. (6.146)

An arbitrary observer u will measure:

(i) an energy density

E(u)=T

αβ

8π

(E(u)

+ B(u)

); (6.147)

(ii) a momentum density (Poynting vector)

P(u)

= −P (u)

4π

[E(u) ×

B(u)]

; (6.148)

(iii) a uniform pressure

p(u)=

Tr T (u)=

E(u). (6.149)

114 Local measurements

Observers U

(em)

who measure a vanishing Poynting vector

Given an electromagnetic ﬁeld and a general observer u, the Poynting vector and

the energy density are given by

4πP(u)=E(u) ×

B(u)=

[

Z(u) ×

Z(u)], (6.150)

4πE(u)=

[E(u)

+ B(u)

|Z(u)|

. (6.151)

It is well known (see exercise 20.6 of Misner, Thorne, and Wheeler, 1973) that

observers exist who see a vanishing Poynting vector or, equivalently, electric and

magnetic ﬁelds parallel to each other. Let U be an observer in relative motion

with respect to u with

U = γ[u + νˆν(U, u)] = u cosh α +ˆν(U, u)sinhα. (6.152)

With respect to U, the magnitude of the Poynting vector is given by

4π||P(U)|| = E(U) ×

B(U )



ˆν(U, u) · (

Z(u) ×

Z(u))



cosh 2α + i |Z(u)|

sinh 2α



|Z(u)|

|cosh 2α tanh 2α

(em)

− sinh 2α|

=4πE(u)

| sinh 2(α −α

(em)

cosh 2α

(em)

, (6.153)

where we have deﬁned

tanh 2α

(em)

= i

ˆν(U, u) · (

Z(u) ×Z(u))

|Z(u)|

ˆν(U, u) · (E(u) ×

B(u))

E(u)

+ B(u)

. (6.154)

When α = α

(em)

, i.e. selecting U as a particular observer U

(em)

,wehave

||P(U

(em)

)|| = 0. With respect to U

(em)

, the electromagnetic energy density takes

a minimum value. In fact, the electromagnetic energy density measured by U is

in general

4πE(U )=

|Z(u)|

[cosh 2α −sinh 2α tanh 2α

(em)

]

=4πE(u)

cosh 2(α −α

(em)

)

cosh 2α

(em)

. (6.155)

Therefore, when α = α

(em)

, E(U

(em)

) takes a minimum value equal to

E(U

(em)

E(u)

cosh 2α

(em)

. (6.156)

6.13 Gravitoelectromagnetism 115

6.12 Time-plus-space (1 + 3) form of Maxwell’s equations

The electromagnetic or Faraday 2-form F is the exterior derivative of a 4-potential

1-form



= dA = u



∧ E(u)



∗

(u)

B(u)



. (6.157)

The splitting of Maxwell’s equations



=0,

∗

F =4πJ (6.158)

leads to

div

B(u)+2ω(u) ·

E(u)=0, (6.159)

curl

E(u)+a(u) ×

E(u)=−[£(u)

+Θ(u)]B(u), (6.160)

div

E(u) −2ω(u) ·

B(u)=4πρ(u), (6.161)

curl

B(u)+a(u) ×

B(u) − [£(u)

+Θ(u)]E(u)=4πJ(u), (6.162)

where J = ρ(u)u+J(u) is the splitting of the 4-current. This shows that Maxwell’s

equations in their traditional form hold only in an inertial frame where ω(u), a(u),

and θ(u) vanish.

6.13 Gravitoelectromagnetism

Consider a unit mass charged particle in motion in a given space-time. The

equation of motion (6.77),

(fw,U,u)

p(U, u)

dτ

(U,u)

= F (U, u)+F

(G)

(fw,U,u)

, (6.163)

becomes

(fw,U,u)

p(U, u)

dτ

(U,u)

= e[E(u)+ν(U, u) ×

B(u)] +

−γ[a(u)+ω(u) ×

ν(U, u)

+ θ(u)

ν(U, u)], (6.164)

where F (U, u) has been replaced by the Lorentz force (6.135) (rescaled by a γ

factor because on the left-hand side the temporal derivative involves the relative

standard time and not the proper time as a parameter along the particle’s world

line), and the gravitational force F

(G)

(fw,U,u)

is given by (3.156), namely

(G)

(fw,U,u)

= −γ[a(u)+ω(u) ×

ν(U, u)+θ(u) ν(U, u)]. (6.165)

Comparing these two forces leads to the identiﬁcation of a gravitoelectric ﬁeld

(u)=−a(u) (6.166)

116 Local measurements

and a gravitomagnetic ﬁeld

(u)=ω(u), (6.167)

so that

(G)

(fw,U,u)

= γ[E

(u)+ν(U, u) ×

(u)+θ(u) ν(U, u)]. (6.168)

Therefore, apart from the gamma factor in the gravitational force and the con-

tribution of the expansion tensor θ(u), the Lorentz force is closely similar to the

gravitational force. This analogy is what we call gravitoelectromagnetism,even

though the gravitational and electromagnetic interactions remain distinct.

It is worth noting that (1) no linear approximation of the gravitational ﬁeld

has been made here in order to introduce gravitoelectromagnetism (Jantzen,

Carini, and Bini, 1992); (2) the analogy between gravity and electromagnetism,

developed here starting from the analysis of test particle motion, could be pursued

similarly studying ﬂuid or ﬁeld motions.

6.14 Physical properties of ﬂuids

A physical observer who studies the behavior of a relativistic ﬂuid which is in

general away from thermodynamic equilibrium must specify in his own rest frame

the parameters which invariantly characterize the ﬂuid and determine its evolu-

tion. For this purpose one would need a full thermodynamic treatment (Israel,

1963), which is well beyond the scope of this book. We shall instead outline some

of the results in the case of a simple ﬂuid, i.e. when the deviations from local

equilibrium are small and the self-gravity of the ﬂuid is neglected.

If T is the energy-momentum tensor of the ﬂuid and u is the vector ﬁeld

tangent to a congruence of observers, the following decomposition holds for each

ﬂuid element (Ellis, 1971; Ellis and van Elst, 1998):

T = T (u)+u ⊗ q(u)+q(u) ⊗ u + ρ(u)u ⊗ u, (6.169)

where

T (u)

αβ

≡ P (u)

P (u)

ρσ

, (6.170)

q(u)

≡−P (u)

ρσ

, (6.171)

ρ(u) ≡ T

ρσ

. (6.172)

An insight into the physical interpretation of these quantities arises from the def-

inition of the 4-momentum of an extended body. If we choose a space-like hyper-

space Σ in such a way that its unit normal is parallel to u, then the quantities

= −T

(6.173)

6.14 Physical properties of ﬂuids 117

turn out to be the components of the 4-momentum density of the ﬂuid. With

respect to the observer u this 4-vector can then be decomposed as

= P (u)

+ T (u)

= −P (u)

+ u

= q(u)

+ ρ(u)u

. (6.174)

The ﬁrst term q(u)

represents the three-dimensional energy ﬂux density and

describes not only the linear momentum of the ﬂuid elements but thermo-

conduction processes such as convection, radiation and heat transfer (Landau and

Lifshitz, 1959; Novikov and Thorne, 1973). The second term ρ(u)u

describes an

energy density current, so its magnitude ρ(u) is just the energy density of the

ﬂuid relative to u. The remaining transverse quantity T (u)in(6.170), being a

symmetric tensor in the three-dimensional LRS

, can be written as the sum of

a trace-free tensor and a trace, as follows:

T (u)

αβ

=[T (u)]

αβ

[Tr T (u)] P (u)

αβ

, (6.175)

where [T (u)]

is the trace-free part of T (u)andTrT (u) is its trace. The quan-

tity [T (u)]

is termed the viscous stress tensor and describes non-isotropic dis-

sipative processes; to the assumed accuracy (namely small deviations from local

equilibrium) the viscous stresses enter the energy-momentum tensor as linear

perturbations to the equilibrium conﬁguration and are given in terms of the ﬂuid

shear, with a coeﬃcient called the shear viscosity. The trace Tr T (u) is only

related to the uniform properties of the ﬂuid; it can be written as

Tr T (u)=3



p(u)+

φ(u)



, (6.176)

where p(u) is the hydrostatic pressure and

φ(u) is a contribution to the pressure

arising from the appearance of a volume (or bulk) viscosity. Finally, we deﬁne

as comoving with the ﬂuid that observer U with respect to whom the quantities

q(U)

describe only processes of thermo-conduction, the ﬂuid elements having

zero momentum (Landau and Lifshitz, 1959). This observer’s four-velocity will

be hereafter be identiﬁed as the 4-velocity of the ﬂuid; the other physical quanti-

ties, density ρ(U), pressure p(U), and internal mechanical stresses T (U), will be

denoted by ρ

, p

,andT

, respectively.

In the case of a perfect ﬂuid, namely when

[T (U )]

=0,

φ =0,q(U)=0,

the energy-momentum tensor becomes

αβ

=[p(U )+ρ(U)]U

+ g

αβ

p(U). (6.177)

118 Local measurements

A perfect ﬂuid is termed dust if, in the comoving frame, p(U) = 0; hence it is

described by the energy-momentum tensor

αβ

= ρ(U)U

. (6.178)

In what follows we will discuss separately the cases of ordinary ﬂuids (i.e. those

without thermal stresses) and ﬂuids with thermal stress, discussing both absolute

and relative dynamics with respect to a general observer.

Ordinary ﬂuids: absolute dynamics

A ﬂuid is termed ordinary when its stresses in the comoving frame, hereafter

referred to as proper stresses, are purely mechanical. In this case its energy-

momentum tensor can be written as

T = ρ

U ⊗ U + T

, T

U =0,ρ

= μ

ˆ

, (6.179)

where U is the 4-velocity ﬁeld of the ﬂuid elements, ρ

includes the matter energy

(μ

) and internal energy (ˆ

), and T

≡T(U ) is the proper mechanical stress

tensor. In the presence of matter and eventually other external ﬁelds the evolution

equations of the ﬂuid are

div T = μ

f, (6.180)

where the space-time divergence operation div is deﬁned in (2.111)andf repre-

sents the action of any external ﬁeld. Let us consider the splitting of (6.180)in

the comoving frame of the ﬂuid. First of all we have

∇

αβ

= ∇

(ρ

+ ρ

∇

+ [div T

]

= μ

, (6.181)

that is

∇

(ρ

U)+ρ

UΘ(U)=μ

, (6.182)

where Θ(U)=∇

and

≡ f −

div T

(6.183)

is the total mechanical force, including both internal and external actions given

respectively by (1/μ

)div T

and f.

Equation (6.182) can then be written in the form

a(U)+U div (ρ

U)=μ

. (6.184)

Projecting (6.184) orthogonally to U then gives

a(U)=μ

P (U)f

≡ μ

(e)

+ f

(i)

), (6.185)

6.14 Physical properties of ﬂuids 119

where the contributions from external and internal forces have been made explicit

according to the following deﬁnitions:

(e)

= P (U )f (6.186)

and

(i)

≡−

P (U)[div T

]

= −

[div

+ a(U) T

] , (6.187)

where we have introduced the spatial divergence operator (see (3.37))

[div

]

= ∇(U)

μα

. (6.188)

The following notation will prove useful:

= −f · U Proper power of external forces

per unit of mass;

(i)

= −U · div T

=Tr[T

θ(U)] Power of internal forces

per unit proper volume;

= μ

− w

(i)

= −μ

· U) Total power per unit

of proper volume. (6.189)

Multiplying Eq. (6.184)byU, one ﬁnds the energy equation

div (ρ

U)=W

. (6.190)

Excluding processes of matter creation or annihilation, one must add to this

equation the conservation of the proper mass of the ﬂuid,

0=div(μ

U)=∇

+ μ

Θ(U). (6.191)

In this case, using the relation ρ

= μ

ˆ

, the energy equation (6.190) becomes

∇

ˆ

; (6.192)

therefore, recalling (6.189), we ﬁnd

Dˆ

dτ

= w

−

(i)

, (6.193)

which represents the First Law of Thermodynamics in the rest frame of the ﬂuid.

Summarizing, the fundamental equations in the rest frame of the ﬂuid are given by

a(U)=μ

(i)

+ f

(e)

) Equation of motion;

Dˆ

dτ

= w

−

(i)

Energy equation. (6.194)

120 Local measurements

The equation of motion can also be cast in the following form. According to

(6.185),

(i)

= −div

− a(U) T

, (6.195)

that is, f

(i)

contains the ﬂuid acceleration a(U). One can collect terms involving

acceleration, rewriting the equation of motion in the form

[ρ

P (U) −T

] a(U)=μ

(e)

− div

, (6.196)

similar to the second law of test particle dynamics.

Ordinary ﬂuids: relative dynamics

We now study ﬂuid dynamics as seen by an observer u not comoving with the

ﬂuid. Splitting the 4-velocity ﬁeld of the ﬂuid in the standard way,

U = γ(U, u)[u + ν(U, u)], (6.197)

the evolution equations with respect to u are obtained by the spatial and temporal

projections of (6.194) with respect to u. Projecting (6.185) ﬁrst orthogonally to

U and then orthogonally to u gives

P (u, U)a(U)=μ

P (u, U)f

. (6.198)

Using relation (6.106), namely

P (u, U)a(U)=−γF

(G)

(fw,U,u)

+ γ

P (u, U, u)a

(fw,U,u)

, (6.199)

with

P (u, U, u) ≡ P (u, U)

P (U, u), (6.200)

one ﬁnds

P (u, U, u)a

(fw,U,u)

= ρ

γF

(G)

(fw,U,u)

+ μ

P (u, U)f

. (6.201)

Let us now deﬁne, following Ferrarese and Bini (2007), the relative energy and

mass densities

ˆμ = ρ

,μ= μ

, (6.202)

where the presence of the square of the gamma factor has a simple explanation in

terms of the Lorentz transformation from the comoving frame U to the observer’s

frame u:oneγ comes from the energy transformation and the other from the

volume transformation. Equation (6.201) then becomes

ˆμa

(fw,U,u)

= P (u, U, u)

−1

[ρ

γF

(G)

(fw,U,u)

+ μ

P (u, U)f

]

= P (u, U, u)

−1

[ρ

γF

(G)

(fw,U,u)

+ μ

P (u)f

− γνW

]. (6.203)

6.14 Physical properties of ﬂuids 121

Let us now introduce the total relative force

μF

(tot)

(fw,U,u)

= μ

P (u)f

+ ρ

γF

(G)

(fw,U,u)

, (6.204)

which contains the contributions of external, internal, and gravitational forces.

The equation of motion becomes

ˆμa

(fw,U,u)

= μP (u, U, u)

−1

(tot)

(fw,U,u)

−

= μP (u, U, u)

−1

(tot)

(fw,U,u)

− Wν, (6.205)

where W

/γ ≡ W represents the total relative power, namely

W =

− w

(i)

= μ ˜w − w

(i)

where we have introduced, from (6.202), the quantities

˜w ≡

(i)

≡

(i)

, (6.206)

which represent the relative power of the external forces per unit of mass ( ˜w)

and of the internal forces (w

(i)

), respectively. Therefore

ˆ

(fw,U,u)



(tot)

(fw,U,u)

− ν(ν ·F

(tot)

(fw,U,u)

)



− ν

, (6.207)

dˆ

dτ

(U,u)

= γ

. (6.208)

Here dτ

(U,u)

= γdτ

is the relative standard time, ˆ =ˆ

,ˆμ =ˆμ, and we have

used the relation

P (u, U, u)

−1

= P (U, u)

−1

P (u, U)

−1

= P (u) − ν(U, u) ⊗ ν(U, u). (6.209)

Transformation law for proper mechanical stresses

The transformation of the spatial (with respect to U ) and symmetric tensor

= T (U) along u and onto LRS

proceeds in a standard way. Let us denote

T (u, U)=P (u, U)T (U), (6.210)

where the projection is understood to be on each index. Acting on (6.210) with

the operator P (u, U)

−1

and recalling that

P (u, U)

−1

P (u, U)=P (U), (6.211)

we obtain

T (U )=P (u, U)

−1

T (u, U). (6.212)

122 Local measurements

If u



denotes another family of observers, then

T (u



,U)=P (u



,U)T (U); (6.213)

hence the transformation law for mechanical stresses follows:

T (u



,U)=[P (u



,U)P (u, U)

−1

]T (u, U). (6.214)

As an example let us evaluate the relative power of internal forces w

(i)

. From its

deﬁnition (see (6.189)) we have

(i)

[T (U )

αβ

∇

]. (6.215)

Recalling that U

= γ [u

+ ν(U, u)

], it follows that

(i)

= T (U)

αβ

∇

+ ν

). (6.216)

Using the transformation law for mechanical stresses (6.214) one then ﬁnds

(i)

= T (U)

αβ

∇

+ ν

)

= T (u, U)

μν

[P (u, U)

−1

]

[P (u, U)

−1

]

[−u

a(u)

− k(u)

αβ

+ ∇

]

= T (u, U)

μσ

[∇

+ ν

∇

]+Tr[T (u, U) θ(u)]

+[ν

T (U, u)] · [−γ

−1

(G)

(fw,U,u)

− 2ν θ(u) − ν(ν · a(u))], (6.217)

where the representation

P (u, U)

−1

= P (u)+u ⊗ ν(U, u) (6.218)

of the mixed projector has been used.

Example: the perfect ﬂuid

In this case T

= p

P (U), where P (U) is the projection operator orthogonal to

U. Using mass conservation we have

(i)

= p

Θ(U). (6.219)

According to the notation previously introduced (see Eqs. (6.189)), we also have

div T

= ∇(U)p

+ p

UΘ(U)+p

a(U), (6.220)

= μ

f −∇(U)p

− p

UΘ(U) − p

a(U), (6.221)

P (U)f

= μ

[f − Uw

] −[∇(U)p

+ p

a(U)], (6.222)

and the evolution equations reduce to

(ρ

+ p

)a(U)=−∇(U)p

+ μ

[f − Uw

dρ

dτ

= −(ρ

+ p

)Θ(U)+μ

. (6.223)