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

Подождите немного. Документ загружается.

Contents ix

9.6 Gravitationally induced time delay 215

9.7 Ray-tracing in Kerr space-time 218

9.8 High-precision astrometry 225

10 Measurements of spinning bodies 231

10.1 Behavior of spin in general space-times 231

10.2 Motion of a test gyroscope in a weak gravitational ﬁeld 237

10.3 Motion of a test gyroscope in Schwarzschild space-time 239

10.4 Motion of a spinning body in Schwarzschild space-time 241

10.5 Motion of a test gyroscope in Kerr space-time 247

10.6 Motion of a spinning body in Kerr space-time 248

10.7 Gravitational waves and the compass of inertia 255

10.8 Motion of an extended body in a gravitational wave space-time 261

Epilogue 265

Exercises 267

References 299

Index 305

Preface

A physical measurement is meaningful only if one identiﬁes in a non-ambiguous

way who is the observer and what is being observed. The same observable can be

the target of more than one observer so we need a suitable algorithm to compare

their measurements. This is the task of the theory of measurement which we

develop here in the framework of general relativity.

Before tackling the formal aspects of the theory, we shall deﬁne what we mean

by observer and measurement and illustrate in more detail the concept which

most aﬀected, at the beginning of the twentieth century, our common way of

thinking, namely the relativity of time.

We then continue on our task with a review of the entire mathematical machin-

ery of the theory of relativity. Indeed, the richness and complexity of that machin-

ery are essential to deﬁne a measurement consistently with the geometrical and

physical environment of the system under consideration.

Most of the material contained in this book is spread throughout the litera-

ture and the topic is so vast that we had to consider only a minor part of it,

concentrating on the general method rather than single applications. These have

been extensively analyzed in Cliﬀord Will’s book (Will, 1981), which remains an

essential milestone in the ﬁeld of experimental gravity. Nevertheless we apologize

for all the references that would have been pertinent but were overlooked.

We acknowledge ﬁnancial support by the Istituto Nazionale di Fisica Nucle-

are, the International Center for Relativistic Astrophysics Network, the Gruppo

Nazionale per la Fisica Matematica of Istituto Nazionale di Alta Matematica,

and the Ministero della Pubblica Istruzione of Italy.

Thanks are due to Christian Cherubini, Andrea Geralico, Giovanni Preti, and

Oldrich Semer´ak for helpful discussions. Particular thanks go to Robert Jantzen

for promoting interest in this ﬁeld. All the blame for any inconsistencies and

errors contained in the book should be addressed to us only.

Fernando de Felice

Physics Department, University of Padova, Italy

Donato Bini

Istituto per le Applicazioni del Calcolo “M. Picone,”

CNR, Rome, Italy

Notation

: The real line.



: The space of the quadruplets of real numbers.

α=0,1,2,3

: A quadruplet of local coordinates.

}: A ﬁeld of bases (frames) for the tangent space.

{ω

}: A ﬁeld of dual bases (dual frames) ω

)=δ

g=g

αβ

⊗ ω

: The metric tensor.

−1

: Inverse metric.

g: Determinant of the metric.

: A tangent vector ﬁeld (with contravariant components).



: The 1-form (with covariant components) g-isomorphic to X.

(left contraction): Contraction of the rightmost contravariant index of the

ﬁrst tensor with the leftmost covariant index of the second tensor, that is

T ]

···

= S

···α

α···

(right contraction): Contraction of the rightmost covariant index of the ﬁrst

tensor with the leftmost contravariant index of the second tensor, that is

T ]

···

= S

···α

α···

(left p-contraction): Contraction of the rightmost p contravariant indices of

the ﬁrst tensor with the leftmost p covariant indices of the second tensor, i.e.

T ≡ S

α...β

...β

...

(right p-contraction): Contraction of the rightmost p covariant indices of the

ﬁrst tensor with the leftmost p contravariant indices of the second tensor, i.e.

T ≡ S

α...

...β

...





-tensor: A tensor r-times contravariant and s-times covariant.

[α

...α

]: Antisymmetrization of the p indices.

(α

...α

): Symmetrization of the p indices.

Notation xiii

[ALT S]

...α

= S

[α

...α

]

[SYM S]

...α

= S

(α

...α

)

(·): γ-component of a frame derivative.

∇: Covariant derivative.

∇

: α-component of the covariant derivative relative to the frame {e



...α

= 

[α

...α

]

: Levi-Civita alternating symbol.

...α

= g

1/2



...α

; η

...α

= −g

−1/2



...α

: The unit volume 4-form.

...α

...β

= 

...α



...β

= −η

...α

...β

: Generalized Kronecker delta.

[

∗

p+1

...α

p+1

...α

: Hodge dual of S

...α

“·”: Scalar g-product, i.e. u ·v = g(u, v)=g

αβ

for any pair of vectors (u, v).

∧: The exterior or wedge product, i.e. u ∧ v = u ⊗ v − v ⊗ u for any pair (u, v)

of vectors or 1-forms.

ˆα

}: An orthonormal frame (tetrad).

: Absolute derivative along a curve γ with parameter s, i.e. D/ds = ∇

˙γ

a(u): Acceleration vector of the world line with tangent vector ﬁeld u,

i.e. a(u)=∇

(fw,u)

: The Fermi-Walker derivative along the curve with parameter s. For any

vector ﬁeld X:

(fw,u)

± [a(u)(u ·X) −u(a(u) · X)].

: The congruence of curves with tangent ﬁeld X.

ω(X): The vorticity tensor of the congruence C

(the same symbol also denotes

the vorticity vector).

θ(X): The expansion tensor of the congruence C

Θ(X)=Trθ(X): The trace of the expansion tensor of the congruence C

: Lie derivative along the congruence C

αβ

: Structure functions of a given frame.

...α

= p!ω

[α

⊗···⊗ω

]

: The dual basis tensor of a space of p-forms.

δT =

∗

T ]: Divergence of a p-form T.

(dR)

= δd + dδ: de Rham operator.

LRS

: Local rest space of u.

P (u): u-spatial projector operator which generates LRS

xiv Notation

T (u): u-temporal projector operator which generates the time axis of u.

[P (u)S] ≡ S(u): Total u-spatial projection of a tensor S such that

[S(u)]

...

= P (u)

···P (u)

···S

...

[£(u)

]: u-spatially projected Lie derivative. For any tensor T it is

[£(u)

T ]

α...

β...

= P (u)

···P (u)

···[£

T ]

...

ρ...

∇(u)

lie

= £(u)

: u-spatial Lie temporal derivative.

∇(u)=P (u)∇: u-spatially projected covariant derivative.

P (u)

(fw,X)

: u-spatially projected Fermi-Walker derivative along a curve with

unit tangent vector X.

d(u)=P (u)d: u-spatially projected exterior derivative.

“·

”: u-spatial inner product, i.e. X ·

Y = P (u)

αβ

“×

”:u-spatial cross product, i.e. [X ×

Y ]

= η(u)

ρσ

η(u)

ρσ

= u

βα

ρσ

: u-spatial 4-volume.

grad

= ∇(u): u-spatial gradient.

curl

= ∇(u)×

: u-spatial curl.

div

= ∇(u)·

: u-spatial divergence.

Scurl

: Symmetrized curl

, i.e. [Scurl

αβ

= η(u)

γδ(α

∇(u)

β)

(fw)

: Fermi-Walker rotation coeﬃcients, i.e. C

(fw)

= e

·∇

(lie)

: Lie rotation coeﬃcients, i.e. C

(lie)

= ω

(£(u)

∇(u)

(fw)

= P (u)∇

: u-spatial Fermi-Walker temporal derivative.

∇(u)

(tem)

. ≡∇(u)

(fw)

or ∇(u)

(lie)

ν(U, u): Relative spatial velocity of U with respect to u.

ν(u, U ): Relative spatial velocity of u with respect to U .

γ(U, u)=γ(u, U )=γ: Lorentz factor of the two observers u and U.

ˆν(u, U ): Unitary relative velocity vector of u with respect to U.

ζ: Angular velocity.

ω(k, u): Frequency of the light ray k with respect to the observer u.

||ν(U, u)|| = ||ν(u, U )|| = ν: Magnitude of the relative velocity of the two

observers u and U.

Notation xv

B(U, u): Relative boost from u to U.

P (U, u)=P (U)P (u): Mixed projector operator from LRS

into LRS

(lrs)

(U, u)=P (U )B(U, u)P (u): Boost from LRS

into LRS

(lrs)

(U, u)

−1

= B

(lrs)

(u, U): Inverse boost from LRS

to LRS

(lrs)

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

−1

(lrs)

(U, u).

(lie,U)

dτ

=[U, X]: Lie derivative of X along C

(U, u)

: Relative standard time parameter, i.e. dτ

(U, u)

= γ(U, u)dτ



(U, u)

: Relative standard length parameter, i.e. d

(U, u)

= γ(U, u)||ν(U, u)||dτ

(fw,U,u)

dτ

(U, u)

= P (u)

dτ

(U, u)

: Projected absolute covariant derivative along U.

(fw,U,u)

= P (u)

Dν(U,u)

dτ

(U,u)

: Relative acceleration of U with respect to u.

(∇X)

αβ

≡∇

Physical dimensions

We are using geometrized units with G =1=c, G and c being Newton’s gravi-

tational constant and the speed of light in vacuum, respectively. Symbols are as

they appear in the text; the reader is advised that more than one symbol may

be used for the same item and conversely the same symbol may refer to diﬀerent

items. Reference is made to the observers (U, u).

Time t → [L]

Space r, x, y, z → [L]

Mass M,m,μ

→ [L]

Angular velocity ζ → [L]

−1

Energy E → [L]

Speciﬁc energy

(in units of mc

) E,γ → [L]

Speciﬁc angular momentum

(in units of mc) L, λ, Λ, → [L]

of a rotating source (Kerr) a → [L]

Spin S → [L]

Speciﬁc spin S/m → [L]

4-velocity U, u → [L]

Relative velocity ν(U, u) → [L]

Force F → [L]

Acceleration a(U) → [L]

−1

Expansion Θ(U) → [L]

−1

xvi Notation

Vorticity (ω(U )

αβ

ω(U)

αβ

)

1/2

→ [L]

−1

Electric charge Q, e → [L]

Space-time curvature (R

αβγδ

)

1/2

→ [L]

−2

Electric ﬁeld E(U) → [L]

−1

Magnetic ﬁeld B(U ) → [L]

−1

Frequency ω(k, u) → [L]

−1

Gravitational wave amplitude h

+,×

→ [L]

Strain S(U) → [L]

−2

Conversion factors

We list here conversion factors from conventional CGS to geometrized units. For

convenience we denote the quantities in CGS units with a tilde (∼).

Name CGS units Geometrized units

Mass

M M =

Electric charge

QQ=



4π

Velocity ˜vν=

˜v

Acceleration ˜aa=

˜a

Force

FF=

Electric ﬁeld

EE=



4π

Magnetic ﬁeld

HH=



4π

Energy

EE=

Speciﬁc energy

≡ γ =

Angular momentum

LL=

Angular momentum in units of

λ =

Introduction

A physical measurement requires a collection of devices such as a clock, a

theodolite, a counter, a light gun, and so on. The operational control of this

instrumentation is exercised by the observer, who decides what to measure, how

to perform a measurement, and how to interpret the results. The observer’s labo-

ratory covers a ﬁnite spatial volume and the measurements last for a ﬁnite interval

of time so we can deﬁne as the measurement’s domain the space-time region in

which a process of measurement takes place. If the background curvature can be

neglected, then the measurements will not suﬀer from curvature eﬀects and will

then be termed local. On the contrary, if the curvature is strong enough that it

cannot be neglected over the measurement’s domain, the response of the instru-

ments will depend on the position therein and therefore they require a careful

calibration to correct for curvature perturbations. In this case the measurements

carrying a signature of the curvature will be termed non-local.

1.1 Observers and physical measurements

A laboratory is mathematically modeled by a family of non-intersecting time-like

curves having u as tangent vector ﬁeld and denoted by C

; this family is also

termed the congruence. Each curve of the congruence represents the history of

a point in the laboratory. We choose the parameter τ on the curves of C

as to make the tangent vector ﬁeld u unitary; this choice is always possible for

non-null curves. Let Σ be a space-like three-dimensional section of C

spanned

by the curves which cross a selected curve γ

∗

of the congruence orthogonally. The

concepts of unitarity and orthogonality are relative to the assumed background

metric. The curve γ

∗

will be termed the ﬁducial curve of the congruence and

referred to as the world line of the observer. Let the point of intersection of Σ

with γ

∗

be γ

∗

(τ); as τ varies continuously over γ

∗

, the section Σ spans a four-

dimensional volume which represents the space-time history of the observer’s

laboratory. Whenever we limit the extension of Σ to a range much smaller than

2 Introduction

the average radius of its induced curvature, we can identify C

with the single

curve γ

∗

and Σ with the point γ

∗

(τ). Any time-like curve γ with tangent vector

u can then be identiﬁed as the world line of an observer, which will be referred to

as “the observer u.” If the parameter τ on γ is such as to make the tangent vector

unitary, then its physical meaning is that of the proper time of the observer u,

i.e. the time read on his clock in units of the speed of light in vacuum.

This concept of observer, however, needs to be specialized further, deﬁning a

reference frame adapted to him. A reference frame is deﬁned by a clock which

marks the time as a parameter on γ, as already noted, and by a spatial frame

made of three space-like directions identiﬁed at each point on γ by space-like

curves stemming orthogonally from it. While the time direction is uniquely ﬁxed

by the vector ﬁeld u, the spatial directions are deﬁned up to spatial rotations,

i.e. transformations which do not change u; obviously there are inﬁnitely many

such spatial perspectives.

The result of a physical measurement is mathematically described by a scalar,

a quantity which is invariant under general coordinate transformations. A scalar

quantity, however, is not necessarily a physical measurement. The latter, in fact,

needs to be deﬁned with respect to an observer and in particular to one of the

inﬁnitely many spatial frames adapted to him. The aim of the relativistic theory

of measurement is to enable one to devise, out of the tensorial representation of

a physical system and with respect to a given frame, those scalars which describe

speciﬁc properties of the system.

The measurements are in general observer-dependent so, as stated, a criterion

should also be given for comparing measurements made by diﬀerent observers.

A basic role in this procedure of comparison is played by the Lorentz group of

transformations. A measurement which is observer-independent is termed Lorentz

invariant. Lorentz invariant measurements are of key importance in physics.

1.2 Interpretation of physical measurements

The description of a physical system depends both on the observer and on the

chosen frame of reference. In most cases the result of a measurement is aﬀected

by contributions from the background curvature and from the peculiarity of the

reference frame. As long as it is not possible to discriminate among them, a

measurement remains plagued by an intrinsic ambiguity. We shall present a few

examples where this situation arises and discuss possible ways out. The most

important among the observer-dependent measurements is that of time intervals.

Basic to Einstein’s theory of relativity is the relativity of time. Hence we shall

illustrate this concept ﬁrst, dealing with inertial frames for the sake of clarity.

1.3 Clock synchronization and relativity of time

The theory of special relativity, formally issued in 1905 (Einstein, 1905), presup-

poses that inertial observers are fully equivalent in describing physical laws. This