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

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6.7 Measurements of acceleration 103

Therefore

P (u, U)a(U)=P (u)∇

= P (u) {γ∇

[u + ν(U, u)] + [u + ν(U, u)]∇

γ}

= γP(u)∇

[u + ν(U, u)] + ν(U, u)∇

γ, (6.71)

that is

P (u, U)a(U)=γP(u)

dτ

+ P (u)

D(γν(U, u))

dτ

= γ

(fw,U,u)

dτ

(fw,U,u)

p(U, u)

dτ

, (6.72)

where p(U, u)=γν(U, u) is the relative linear momentum of the particle per unit

mass, with respect to u. We have already deﬁned in (3.156) the gravitational

force (per unit mass) as

P (u)

dτ

= −F

(G)

(fw,U,u)

; (6.73)

hence we have in the local rest frame of u:

P (u, U)a(U)=−γF

(G)

(fw,U,u)

p(U, u)

dτ

. (6.74)

Let us deﬁne

P (u, U)f(U) ≡ γF(U, u), (6.75)

then recalling (6.67)wehave

− γF

(G)

(fw,U,u)

p(U, u)

dτ

= γF(U, u), (6.76)

so that, from (3.151), we obtain

(fw,U,u)

p(U, u)

dτ

(U,u)

= F (U, u)+F

(G)

(fw,U,u)

. (6.77)

This is the force equation in a Newtonian-like form. Finally, scalar multiplication

of both sides of (6.77)byν(U, u) gives

ν(U, u) ·

(fw,U,u)

p(U, u)

dτ

(U,u)

= ν(U, u) ·[F (U, u)+F

(G)

(fw,U,u)

]. (6.78)

Recalling that p(U, u)=γν(U, u), we have

ν(U, u) ·

(fw,U,u)

(γν(U, u))

dτ

(U,u)

dγ

dτ

(U,u)

+ γ



ν(U, u) ·

(fw,U,u)

ν(U, u)

dτ

(U,u)



dγ

dτ

(U,u)





dγ

dτ

(U,u)

, (6.79)

104 Local measurements

where we have used the relation

dγ

dτ

(U,u)

= γ



ν(U, u) ·

(fw,U,u)

ν(U, u)

dτ

(U,u)



. (6.80)

Therefore, we have

dE(U, u)

dτ

(U,u)

= ν(U, u) ·[F (U, u)+F

(G)

(fw,U,u)

], (6.81)

where E(U, u)=γ(U, u) ≡ γ is the energy of the particle per unit of mass.

Equation (6.81)isthepower equation in a Newtonian-like form.

Longitudinal-tranverse splitting of the force equation

The measurement of the acceleration of a test particle in the presence of an exter-

nal force permitted a Newtonian-like representation of the equations of motion,

as shown by (6.77) and (6.81). One can complete this analysis by considering a

further splitting of these equations along directions parallel (longitudinal) and

orthogonal (transverse) to that of the velocity of the particle relative to the

observer u. A key tool for this study is the use of the relative Frenet-Serret

frames introduced in Chapter 4.

The transverse splitting of the relative acceleration deﬁnes the relative cen-

tripetal acceleration. This splitting is accomplished by projecting (6.77)onto

the relative Frenet-Serret frame {ˆν(U, u), ˆη

(fw,U,u)

}, as discussed in

Eqs. (4.39), (4.40), and (4.41).

Let ||p(U, u)|| = γ||ν(U, u)|| be the magnitude of the speciﬁc (i.e. per unit mass)

spatial momentum of U as seen by u,sothat

p(U, u)=||p(U, u)||ˆν(U, u), (6.82)

while the gamma factor is the corresponding speciﬁc energy E(U, u)=γ. They

satisfy the identity

E(U, u)

−||p(U, u)||

=1. (6.83)

Then (6.77) takes the form

F (U, u)+F

(G)

(fw,U,u)

d||p(U, u)||

dτ

(U,u)

ˆν(U, u)+||p(U, u)||

(fw,U,u)

ˆν(U, u)

(U,u)

d||p(U, u)||

dτ

(U,u)

ˆν(U, u)

+ γν(U, u)

(fw,U,u)

ˆη

(fw,U,u)

, (6.84)

where (4.39) has been used.

6.7 Measurements of acceleration 105

The relative Frenet-Serret components of this equation, i.e. the projections on

the axes of the triad {ˆν(U, u), ˆη

(fw,U,u)

}, are given by

dE(U, u)

d

(U,u)

=[F (U, u)+F

(G)

(fw,U,u)

] · ˆν(U, u),

γν(U, u)

(fw,U,u)

=[F (U, u)+F

(G)

(fw,U,u)

] · ˆη

(fw,U,u)

0=[F (U, u)+F

(G)

(fw,U,u)

] ·

(fw,U,u)

, (6.85)

where d

(U,u)

= ντ

(U,u)

and

d||p(U, u)||

dτ

(U,u)

E(U, u)

||p(U, u)||

dE(U, u)

dτ

(U,u)

dE(U, u)

dτ

(U,u)

dE(U, u)

d

(U,u)

. (6.86)

This equality follows from the identity (6.83), and has been used to express the

longitudinal acceleration term in (6.84) as shown by (6.85)

Note that if U is tangent to a geodesic, F (U, u) = 0 and Eqs. (6.85) imply

dE(U, u)

d

(U,u)

= F

(G)

(fw,U,u)

· ˆν(U, u),

γν(U, u)

(fw,U,u)

= F

(G)

(fw,U,u)

· ˆη

(fw,U,u)

0=F

(G)

(fw,U,u)

. (6.87)

On the other hand, the force equation may also be considered from the point

of view of the comoving Frenet-Serret frame {

V(u, U),

(fw,u,U )

} intro-

duced in Chapter 4, Eqs. (4.46).

By applying the derivative D/dτ

to the representation of u,

u = γ(U + ν(u, U )) = γ(U − ν

V(u, U)) (6.88)

and solving for a(U)=f(U ), one obtains

a(U)=

dτ

= P (U )

dτ

= P (U )

dτ



−1

u −ν(u, U)



= P (U )

D(γ

−1

dτ

+ P (U)

dτ



V(u, U)



= −γ

−2

Dγ

dτ

P (U)u + γ

−1

P (U)

dτ

Dν

dτ

V(u, U)+νP(U)

V(u, U)

dτ

. (6.89)

Using (6.88), we now ﬁnd that

P (U)u = −γν

V(u, U), (6.90)

106 Local measurements

so that, recalling (3.156) for the deﬁnition of the gravitational force, we have for

the previous relation

f(U)=−γν

dν

dτ

(−γν

V(u, U)) − γ

−1

P (U, u)F

(G)

(fw,U,u)

dν

dτ

V(u, U)+νP(U)

V(u, U)

dτ

=(1+γ

)

dν

dτ

V(u, U) − γ

−1

P (U, u)F

(G)

(fw,U,u)

+ νP(U)

V(u, U)

dτ

= γ

dν

dτ

V(u, U) − γ

−1

P (U, u)F

(G)

(fw,U,u)

+ νP(U)

V(u, U)

dτ

= γ

−1

d(γν)

dτ

V(u, U) − γ

−1

P (U, u)F

(G)

(fw,U,u)

+ νP(U)

V(u, U)

dτ

= γ

−1

d||p(u, U)||

dτ

V(u, U)+γν

(fw,u,U )

−γ

−1

P (U, u)F

(G)

(fw,U,u)

. (6.91)

Rearranging terms in this equation leads to the suggestive form

d||p(u, U)||

dτ

(U,u)

V(u, U)=f(U)+F

(G)

(fw,u,U)

−γν

(fw,u,U )

, (6.92)

where

(G)

(fw,u,U)

= γ

−1

P (U, u)F

(G)

(fw,U,u)

= −γ

−1

P (U)

dτ

= −P (U )

dτ

(U,u)

. (6.93)

The last term on the right-hand side of (6.92), being proportional to the derivative

of the unit spatial velocity

V(u, U)ofu with respect to U , is called the generalized

centrifugal force; it is measured by the particle U itself. Its non-relativistic limit

in ﬂat space-time leads to the familiar centrifugal force relative to the usual

family of inertial observers. The comoving relative Frenet-Serret decomposition

of (6.92)is

dE(U, u)

d

(U,u)

=[f (U)+F

(G)

(fw,u,U )

] ·

V(u, U),

γν

(fw,u,U )

=[f (U)+F

(G)

(fw,u,U )

] ·

(fw,u,U )

0=[f (U)+F

(G)

(fw,u,U )

] ·

(fw,u,U )

. (6.94)

These equations are valid as long as ν belongs to the open interval (0, 1) and with

a slight modiﬁcation when ν = 1. When ν = 0, the spatial arc length parameter-

ization d

(U,u)

= γνdτ

is singular because the particle path in LRS

collapses

to a point. However, since Eqs. (6.94) are just projections along the comoving

Frenet-Serret triad of the space-time force equation, they continue to hold in

6.7 Measurements of acceleration 107

the limit ν → 0. One may use the terminology comoving relatively straight and

comoving relatively ﬂat for those world lines for which the comoving relative cur-

vature and torsion respectively vanish. By solving Eq. (6.84) for the gravitational

force

(G)

(fw,U,u)

= γν

ˆη

(fw,U,u)

− F (U, u)+

d||p(U, u)||

dτ

(U,u)

ˆν(U, u) (6.95)

and inserting the result into Eq. (4.47), that is

(fw,u,U )

= γk

(fw,U,u)

ˆη

(fw,U,u)

+ˆν(U, u) ×

[ˆν(U, u) ×

(G)

(fw,U,u)

], (6.96)

one ﬁnds the relation

(fw,u,U )

= γ

−1

(fw,U,u)

ˆη

(fw,U,u)

− ˆν

(U,u)

[ˆν

(U,u)

F (U, u)]. (6.97)

When the curve is a geodesic, that is when F (U, u) = 0, the two curvatures

(fw,u,U )

and k

(fw,U,u)

only diﬀer by a gamma factor and the two directions

(fw,u,U )

and ˆη

(fw,U,u)

coincide. In fact, from (6.97) we ﬁnd

(fw,u,U )

= γ

−1

(fw,U,u)

ˆη

(fw,U,u)

, (6.98)

so that

(fw,u,U )

= γ

−1

(fw,U,u)

. (6.99)

Thus, for a geodesic, the notions of relative Fermi-Walker straight (as stated

following (4.42)) and comoving relatively straight world lines agree with each

other.

A relationship between the relative and comoving torsion may be established

by diﬀerentiating Eq. (4.47) along U and using the last comoving relative Frenet-

Serret relation (4.46). After some algebra one ﬁnds

γνK

(fw,u,U )

= γ

νk

(fw,U,u)

− F

(G)

(fw,U,u)

· ˆη

(fw,U,u)

]

d(γk

(fw,U,u)

)

dτ

(G)

(fw,U,u)

]

+ K

(fw,u,U )

·∇

[ˆν(U, u) ×

(G)

(fw,U,u)

]. (6.100)

In the special case of geodesics, this reduces to

(fw,u,U )

= τ

(fw,U,u)

, (6.101)

so the notions of relative Fermi-Walker ﬂatness and comoving relative Fermi-

Walker ﬂatness also agree.

108 Local measurements

6.8 Acceleration change under observer transformations

Consider two accelerated time-like world lines with unit tangent vectors U and u.

We shall express the acceleration of U in terms of that of u. By deﬁnition we have

a(U)=P (U)∇

U = γP(U)[∇

u + ∇

ν(U, u)]

= γP(U)[∇

u + P (u)∇

ν(U, u) − u(u ·∇

ν(U, u))]

= γP(U)[∇

u + P (u)∇

ν(U, u)+ u(ν(U, u) ·∇

u)]. (6.102)

Recalling (3.156) and (3.161), namely

dτ

= −F

(G)

(fw,U,u)

,P(u)

dτ

ν(U, u)=γa

(fw,U,u)

, (6.103)

relation (6.102) becomes

a(U)=γP(U )



−F

(G)

(fw,U,u)

+ γa

(fw,U,u)

− u(ν(U, u) · F

(G)

(fw,U,u)

)



= −γP(U)[P (u)+u ⊗ ν(U, u)] F

(G)

(fw,U,u)

+ γ

P (U, u)a

(fw,U,u)

. (6.104)

But, from (3.121)

it follows that

P (u)+u ⊗ ν(U, u)=P (u, U)

−1

, (6.105)

and hence the ﬁnal result from (3.156) is given by

a(U)=−γP(u, U)

−1

(G)

(fw,U,u)

+ γ

P (U, u)a

(fw,U,u)

= γ



P (u, U)

−1

[a(u)+ω(u) ×

ν(U, u)+θ(u) ν(U, u)]

+ P(U, u)a

(fw,U,u)



. (6.106)

6.9 Kinematical tensor change under observer

transformations

We now consider two time-like congruences of curves, C

and C

; our aim is to

express the kinematical tensor of one congruence in terms of the other. From the

deﬁnition, the kinematical tensor of C

, k(U)=−[P (U)∇U ], can be written as

k(U)

= −P (U )

P (U)

[∇U]

= −P (U )

P (U)

∇

= −γP(U)

P (U)

∇

−γ[∇(U)ν(U, u)]

. (6.107)

6.9 Kinematical tensor change under observer transformations 109

We evaluate the term P (U)∇u as follows:

[P (U)∇u]

= P (U )

P (U)

∇

= P (U )

P (U)

[−a(u)

− k(u)

]. (6.108)

Taking into account that

0=P (U)

= γP(U)

+ ν(U, u)

], (6.109)

we have

P (U)

= −P (U )

ν(U, u)

. (6.110)

Therefore

[P (U)∇u]

= P (U )

P (U)

[a(u)

ν(U, u)

− k(u)

]

= P (U, u)

P (U, u)

[a(u)

ν(U, u)

− k(u)

] , (6.111)

or in index-free notation

P (U)∇u = −P (U, u)[k(u) − a(u) ⊗ ν(U, u)]. (6.112)

We can then complete our task, obtaining

k(U)=γP(U, u)[k(u) − a(u) ⊗ν(U, u)]

−γ∇(U)ν(U, u). (6.113)

Since ω(U )



=ALT[k(U)



], we have the transformation law for the vorticity

2-form,

ω(U)



= γALT

P (U, u)[k(u)



− a(u)



⊗ ν(U, u)



]

γd(U)ν(U, u)



= γP(U, u)[ω(u)



−

a(u) ∧ν(U, u)]

γd(U)ν(U, u)



. (6.114)

We shall now derive the analogous expression for the vorticity vector, namely

ω(U)=γ

P (u, U)

−1

[ω(u)+

ν(U, u) ×

a(u)]

γ curl

ν(U, u). (6.115)

Let us write (6.114)intheform

ω(U)



−

γd(U)ν(U, u)



≡ P (U, u)X, (6.116)

where



= γ



ω(u) −

a(u) ∧ν(U, u)





. (6.117)

110 Local measurements

Next, contract both sides of (6.116) with (1/2)η(U). The left-hand side becomes

η(U)

αβγ



ω(U)

αβ

−

2∇(U)

[α

ν(U, u)

β]



, (6.118)

that is

ω(U)

−

[curl

ν(U, u)]

. (6.119)

To deduce the eﬀect of contracting the right-hand side of (6.116)by(1/2)η(U),

namely

η(U)P (U, u)X,

let us ﬁrst recall, from (3.20), that

η(U)

βγδ

= U

αβγδ

= −2U



[α

η(u)

β]γδ

+ u

[γ

η(u)

δ]αβ



= γ [η(u)

βγδ

+ ν

η(u)

αγδ

+ u

η(u)

αδβ

− u

η(u)

αγβ

)] . (6.120)

Then let us recall that

P (U, u)

= P (u)

+ γU

(6.121)

implies, for any antisymmetric 2-tensor X ∈LRS

⊗LRS

, the following relation:

[P (U, u)X]

αβ

= P (U, u)

P (U, u)

μν

= X

αβ

+ γU

ασ

− γU

βσ

=[X + γU ∧(X ν)]

αβ

. (6.122)

From the latter equation we deduce that

η(U)

γαβ

[P (U, u)X]

αβ

= η(U )

γαβ

αβ

, (6.123)

because terms along U vanish after contraction with η(U ). Using (6.120)wenow

ﬁnd that

η(U)

γαβ

[P (U, u)X]

αβ

= γ[η(u)

γαβ

− ν

η(u)

μβα

αβ

, (6.124)

that is

∗

(U)

[P (U, u)X]=γ[

∗

(u)

X + u ⊗(ν

∗

(u)

X)]

= γ[P (u)+u ⊗ ν]

∗

(u)

= γP(u, U)

−1 ∗

(u)

X. (6.125)

Since we have

∗

(u)

X = γ



ω(u) −

a(u) ×

ν(U, u)



, (6.126)

Eq. (6.115) follows.

6.10 Measurements of electric and magnetic ﬁelds 111

Similarly one can obtain the expansion tensor of C

in terms of the kinematical

properties of C

. Since θ(U)



= −SYM [k(U)



]wehave

− θ(U )



= γSYM

P (U, u)[k(u)



− a(u)



⊗ ν(U, u)



]

−γSYM[∇(U)ν(U, u)



]

= −γP(U, u)



θ(u)



[a(u) ⊗ν(U, u)+ν(U, u) ⊗ a(u)]



−γSYM[∇(U)ν(U, u)



]. (6.127)

6.10 Measurements of electric and magnetic ﬁelds

Assume that an observer u moves through an electromagnetic ﬁeld described

by the Faraday 2-form F

αβ

. The observer learns of this electromagnetic ﬁeld

by studying the behavior of a charged particle. Let U be the 4-velocity of the

particle; its trajectory is not a geodesic because of the Lorentz force, hence it has

a 4-acceleration

a(U)

, (6.128)

where e is the particle’s electric charge and μ

its mass. Our aim is to show how

the measured force on the charge is deﬁned in terms of the charge and the four-

dimensional quantities which characterize the observer and the electromagnetic

ﬁeld. Because F

αβ

is a 2-form, we can apply the general splitting formula (3.14),

valid for a p-form. The result is:

αβ

=2u

[α

E(u)

β]

∗

(u)

B(u)]

αβ

, (6.129)

where we have introduced the electric part of F ,

E(u)

≡ F

, (6.130)

and the magnetic part,

B(u)

≡

∗

ραμν

μν

. (6.131)

In coordinate-free notation the deﬁnition of such ﬁelds is

E(u)=F

u, B(u) ≡

∗

(u)

[P (u)F ]. (6.132)

Let us decompose the force term f(U )=(e/μ

)F U into a transverse and a

parallel component relative to u, namely

f(U)=P (u)

f(U)+T (u) f(U). (6.133)

112 Local measurements

The transverse force can be written in coordinate components as

P (u)

f(U)

P (u)

αβ

βμ

P (u)

αβ

E(u)

− u

E(u)

+ η(u)

βμ

B(u)

] γ(u

+ ν(U, u)

)

= −



σπλ

B(u)

+ E(u)

)



. (6.134)

As a result the transverse force term becomes

P (u)f(U)=

γ [E(u)+ν(U, u) ×

B(u)] . (6.135)

The parallel force component T (u)f(U ) can be written more conveniently as

T (u)

f(U)

= −u



f(U)



= −

. (6.136)

Recalling that

= γ[u

+ ν(U, u)

], (6.137)

we can write (6.136)as

T (u)

f(U)

= u

ν(U, u)

E(U)

. (6.138)

The magnitude of this quantity measures the power of the electromagnetic action

on the particle as seen by u. From the representation of F we have, relative to

both the observers u and U,

αβ

=2u

[α

E(u)

β]

∗

(u)

B(u)]

αβ

=2U

[α

E(U)

β]

∗

(U)

B(U )]

αβ

; (6.139)

hence, contracting with U and using the deﬁnitions so far introduced, we obtain

E(U)=γP(u, U)

−1

[E(u)+ν(U, u) ×

B(u)] ,

B(U )=γP(u, U)

−1

[B(u) − ν(U, u) ×

E(u)] . (6.140)

If we split the ﬁelds along directions parallel and transverse to that of the

relative velocity ˆν(U, u) we obtain

E(u)=E



(u)ˆν(U, u)+E

⊥

(u), (6.141)

and similarly for B(u). Hence (6.140) take the more familiar form

(lrs)

(u, U) E(U)]



= E



(u),

(lrs)

(u, U) E(U)]

⊥

= γ[E

⊥

(u)+ν(U, u) ×

⊥

(u)], (6.142)

and

(lrs)

(u, U) B(U )]



= B



(u),

(lrs)

(u, U) B(U )]

⊥

= γ[ B

⊥

(u) −ν(U, u) ×

⊥

(u)]. (6.143)