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Nonlocal Gravity$

Bahram Mashhoon

Print publication date: 2017

Print ISBN-13: 9780198803805

Published to Oxford Scholarship Online: July 2017

DOI: 10.1093/oso/9780198803805.001.0001

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Introduction

Introduction

Chapter:
(p.1) 1 Introduction
Source:
Nonlocal Gravity
Author(s):

Bahram Mashhoon

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198803805.003.0001

Abstract and Keywords

This introductory chapter is mainly about the locality postulate of the standard relativity theory. The fundamental laws of microphysics have been formulated with respect to inertial observers. However, inertial observers do not in fact exist, since actual observers are accelerated. What do accelerated observers measure? Lorentz invariance is extended to accelerated observers by assuming that they are pointwise inertial. That is, an accelerated observer at each instant is equivalent to an otherwise identical momentarily comoving inertial observer. This hypothesis of locality, which underlies the special and general theories of relativity, is described in detail. The locality postulate fits perfectly together with Einstein’s local principle of equivalence to ensure that every observer in a gravitational field is pointwise inertial. When coupled with the hypothesis of locality, Einstein’s principle of equivalence provides a physical basis for a field theory of gravitation that is consistent with local Lorentz invariance.

Keywords:   inertial observer, accelerated observer, hypothesis of locality, Einstein’s principle of equivalence, field theory of gravitation

A basic locality postulate permeates through the standard special and general theories of relativity. The purpose of this initial chapter is to identify the locality assumption and briefly study its physical origin as well as the significant role that it plays in relativity theory.

1.1 Lorentz Invariance

The principle of relativity—namely, the assertion that the laws of microphysics are the same in all inertial frames of reference—refers to the measurements of ideal inertial observers. The transition from Galilean invariance of Newtonian physics to Lorentz invariance marks the beginning of modern relativity theory. Lorentz invariance is the invariance of the fundamental laws of microphysics under the group of passive inhomogeneous Lorentz transformations. Lorentz invariance has firm observational support; therefore, we assume throughout that Lorentz invariance is a fundamental symmetry of nature. The basic laws of microphysics have been formulated with respect to ideal inertial observers, since these are conceived to be free of the various limitations associated with actual observers. Each ideal inertial observer is forever at rest in a global inertial frame of reference, namely, a Cartesian coordinate system that is homogeneous and isotropic in space and time and in which Newton’s fundamental laws of motion are valid.

The global inertial frames of reference are all related to each other by passive inhomogeneous Lorentz transformations of the form

(1.1)
xμ=Λμνxν+bμ,

where an event is characterized by inertial coordinates xμ=(ct,x), (Λμν) is a Lorentz matrix and bμ is a constant vector of spacetime translation. The set of all such events constitutes flat Minkowski spacetime. In our convention, Greek indices run from 0 to 3, while Latin indices run from 1 to 3; moreover, the spacetime metric has signature +2.

The inhomogeneous Lorentz transformations form the Poincaré group, which is the ten-parameter group of isometries of Minkowski spacetime. That is, the Minkowski spacetime interval ds given by

(1.2)
ds2=ηαβdxαdxβ

is preserved under the Poincaré group. Here, ηαβ is the Minkowski metric tensor given by diag(−1, 1, 1, 1), in accordance with our convention regarding metric signature. (p.2) The four-parameter Abelian group of spacetime translations and the six-parameter Lorentz group, which consists of boosts and rotations, are subgroups of the Poincaré group. It follows from eqns (1.1) and (1.2) that

(1.3)
ηαβΛαμΛβν=ημν.

For μ=ν=0, eqn (1.3) can be expressed as

(1.4)
(Λ00)2=1+i=13(Λi0)2.

Moreover, taking the determinant of both sides of eqn (1.3), we find

(1.5)
det(Λαμ)=±1.

Henceforth, we work with the proper orthochronous Lorentz group (Streater and Wightman 1964), whose elements satisfy det(Λαμ)=1 as well as Λ001. Clearly, the Lorentz group contains the identity element Λαμ=δμα.

The determination of temporal and spatial intervals constitutes the most basic measurements of a physical observer. We assume that each inertial observer has access to an ideal clock as well as infinitesimal measuring rods, and carries along its world line an orthonormal tetrad frame (or vierbein), that is, a set of four unit axes that are orthogonal to each other and characterize the observer’s local temporal and spatial axes. For instance, the local axes of the class of fundamental observers at rest in a global inertial frame are parallel to the corresponding global axes and are given by

(1.6)
λμ0ˆ=(1,0,0,0),λμ1ˆ=(0,1,0,0),

(1.7)
λμ2ˆ=(0,0,1,0),λμ3ˆ=(0,0,0,1).

The hatted tetrad indices at an event enumerate the tetrad axes in the tangent space at that event. In particular, λμ0ˆ is the temporal axis of the observer, while λμiˆ, for i=1,2,3, constitutes the observer’s spatial frame. Each inertial observer in Minkowski spacetime belongs to a class of fundamental observers.

For a tetrad frame λμαˆ carried by an arbitrary inertial observer in a global inertial frame, the orthonormality condition takes the form

(1.8)
ημνλμαˆλνβˆ=ηαˆβˆ.

The tetrads that we consider throughout are adapted to the observers under consideration, which have future-directed timelike world lines and employ right-handed spatial frames in conformity with the right-handed Cartesian coordinates of the background space. Therefore, we limit our considerations to tetrads for which det(λμαˆ)=1. Spacetime indices are in general raised and lowered via the spacetime metric tensor gμν, which is equal to ημν in the present case, while the hatted tetrad indices—that is, the local Lorentz indices—are raised and lowered via the Minkowski metric tensor ημˆνˆ.

(p.3) In connection with spacetime measurements, we imagine static inertial observers in a global inertial frame and assume that their clocks are all synchronized; that is, adjacent clocks can be synchronized. Moreover, the adiabatic transport of a clock to another location can be so slow as to have negligible practical impact on synchronization. In a similar way, lengths can be measured in general by placing infinitesimal rods together. Furthermore, it is assumed in general that for physical measurements, inertial observers have access to ideal measuring devices. These are free from the specific practical limitations of laboratory devices that are usually due to the nature of their construction and modes of operation. The measurements of moving inertial observers are related to those at rest via Lorentz invariance, which preserves the causal sequence of events.

An equivalent (“radar") approach to spacetime measurements relies on the transmission and reception of light signals. In this procedure, a static inertial observer O1 sends out a light signal at time t1 to static inertial observer O2. The signal is immediately transponded without delay back to O1 and is received at time t2. If the clocks at O1 and O2 are synchronized, they would both register time t=(t1+t2)/2 at the instant the signal is received at O2. Moreover, the distance between O1 and O2 is D12=c(t2t1)/2. Thus t2t=tt1=D12/c.

The inertial physics that is based on the ideal inertial observers and their tetrad frames has played a significant role in the development of theoretical physics. Inertial physics was originally established by Newton (Cohen 1960).

1.1.1 Inertial observers

Imagine a background global inertial frame in Minkowski spacetime. The ideal inertial observers in this arena are either at rest with local spatial reference frames that are related to the global axes by a constant rotation or move with constant speeds on straight lines from minus infinity to plus infinity and carry constant local reference frames. The fundamental inertial observers are all at rest and carry orthonormal tetrad frames with axes that coincide with the global Cartesian spacetime axes of the background inertial frame of reference.

The translational motion of the observer in spacetime fixes its local temporal axis as well as its spatial frame but only up to an element of the rotation group. Consider, for illustration, a background inertial frame with coordinates (ct,x) in Minkowski spacetime. An inertial observer moves with constant velocity v relative to the background frame. The Lorentz transformation to the rest frame of the moving observer (ct,x) involving a pure boost with no rotation is given by

(1.9)
t=γt+vxc2,

(1.10)
x=x+1v2(γ1)(xv)v+γvt,

where v=|v| and γ‎ is the corresponding Lorentz factor, namely, γ=1/1v2/c2. In the (ct,x) frame, the tetrad of the fundamental inertial observers at rest is given by hμαˆ=δαˆμ. Transforming the local tetrad frame of the moving observer to the (ct,x) (p.4) system via the Lorentz boost matrix that can be simply deduced from eqns (1.9) and (1.10), we find

(1.11)
hμ0ˆ=γ1,vc,

(1.12)
hμiˆ=δiμ+viγc,(γ1)v2v.

Up to a constant rotation, this is the orthonormal tetrad frame of the boosted inertial observer with respect to the background (ct,x) system. For the generalization of this result to curved spacetimes, see Mashhoon and Obukhov (2014).

The temporal axis of the moving inertial observer’s tetrad, hμ0ˆ, is equal to the unit timelike vector that is the 4-velocity of the observer, hμ0ˆ=dxμ/dτ, where the temporal parameter τ‎ is simply related to the invariant spacetime interval along the observer’s world line. That is, ds2=dτ2, where dτ=cdt/γ. We can clearly identify τ‎ with ct, the proper time of the moving inertial observer. The observer’s spatial frame consists, up to a constant rotation, of the three orthogonal unit spacelike axes given by hμiˆ for i=1,2,3.

Ideal inertial observers all have straight world lines. Imagine, for instance, an inertial observer O0 moving along the positive z axis with constant speed v. Let us introduce the rapidity parameter Θ‎0 such that v/c=tanhΘ0; then, eqns (1.11) and (1.12) imply that the orthonormal tetrad frame of O0 is given by

(1.13)
hμ0ˆ=(coshΘ0,0,0,sinhΘ0),hμ1ˆ=(0,1,0,0),

(1.14)
hμ2ˆ=(0,0,1,0),hμ3ˆ=(sinhΘ0,0,0,coshΘ0).

The path of the observer in spacetime is then rectilinear; that is, it follows from the integration of the 4-velocity vector of O0, uμ=dxμ/dτ=hμ0ˆ, that

(1.15)
ct=τcoshΘ0,x=y=0,z=τsinhΘ0,

where the integration constants have been chosen such that at t=0, τ=0 and the observer is at the origin of spatial coordinates.

1.1.2 Examples of uniformly accelerated observers

Realistic observers in a global inertial frame in Minkowski spacetime would all be more or less accelerated. We consider here some examples of uniformly accelerated observers.

Let us first imagine a noninertial observer Oˆ that has the same history as O0 for τ<0, but Oˆ is forced to accelerate uniformly along the positive z direction starting at τ=0, when Oˆ is at the origin of spacetime coordinates. For τ>0, the observer’s orthonormal tetrad frame is then λμαˆ such that

(1.16)
λμ0ˆ=(coshΘ,0,0,sinhΘ)

and

(1.17)
λμ1ˆ=(0,1,0,0),λμ2ˆ=(0,0,1,0),λμ3ˆ=(sinhΘ,0,0,coshΘ).

(p.5) Here,

(1.18)
Θ=Θ0+g0τc2

and g0 is the constant invariant acceleration of Oˆ. The 4-acceleration aμ of Oˆ is

(1.19)
aμ=dλμ0ˆdτ=g0c2λμ3ˆ.

It follows in general from uμuμ=1 that aμuμ=0, so that the acceleration 4-vector is spacelike with aμaμ=g˜2, where g˜(τ)>0 is the magnitude of the translational acceleration with dimensions of (length)−1. For constant linear acceleration, for instance, g˜=|g0|/c2.

In connection with the propagation of the tetrad frame along the world line of Oˆ, let us briefly digress here and discuss a more general situation that involves an observer following a timelike path in an inertial frame in Minkowski spacetime. The observer has 4-velocity uμ=dxμ/dτ and 4-acceleration aμ=duμ/dτ and carries a vector Vμ along its path. One can decompose this vector into its parallel and perpendicular components relative to the path, namely Vμ=Vμ+Vμ, where the component parallel to the curve is Vμ=(uV)uμ and the corresponding perpendicular component is Vμ=(ημν+uμuν)Vν. Here, uV=uμVμ and |uV| is the length of the parallel component of V. It follows that VV=VV+VV, which is reminiscent of the Pythagorean theorem. Let us now suppose that Vμ is so transported that it does not rotate and its length remains constant; that is, along the path we have

(1.20)
d(uV)dτ=0,dVμdτ=0.

These relations mean that while the magnitude of the parallel component of vector Vμ remains constant along the path, the perpendicular component cannot change in the perpendicular direction; otherwise, vector Vμ would rotate. That is, the net variation of the perpendicular component along the path can only be in the direction parallel to the path. In this way, we find from (1.20) that

(1.21)
dVμdτ=(Va)uμ(Vu)aμ,

which is the Fermi–Walker transport law. We note, in particular, that the 4-velocity of the observer uμ satisfies eqn (1.21).

Returning to the case of observer Oˆ, it is straightforward to check that its tetrad frame is Fermi–Walker transported along its world line. The path of the observer for τ>0 is given by

(1.22)
ct=c2g0(sinhΘsinhΘ0),x=y=0,z=c2g0(coshΘcoshΘ0),

which is a hyperbola in the (ct,z) plane. That is, it follows from (1.22) that

(1.23)
(z+c2g0coshΘ0)2(ct+c2g0sinhΘ0)2=c4g02,

where c2/|g0| is the invariant acceleration length associated with Oˆ for τ>0.

(p.6) We note that at any instant of time τ>0, the tetrad frame (1.16)–(1.17) of the hyperbolic observer Oˆ is of the general form of the frame (1.13)–(1.14) with an instantaneous speed c tanhΘ such that |tanhΘ|1 for τ. This corresponds to the fact that the asymptotes of the hyperbola in (1.23) are null lines. Over the period of proper time τ‎ from 0, an external source causing the constant acceleration of Oˆ must provide an infinite amount of energy to sustain the complete hyperbolic motion. To avoid such unphysical situations, we assume that in general the observer’s acceleration is turned on only over time intervals such that the net amount of energy transfer is always finite.

Observers in a laboratory fixed on the Earth in general rotate in space as the Earth rotates about its axis. Moreover, such observers generally refer their measurements to spatial axes that are rigidly attached to the Earth. Thus their spatial frames rotate with the Earth as well. It is therefore interesting to study tetrad frames adapted to such rotating observers.

To describe uniformly rotating observers in Minkowski spacetime, let us first imagine observers O˜ at rest in a global inertial frame with spacetime coordinates (ct,x,y,z). However, these static observers are not inertial since instead of the (x,y,z) axes of the background frame, they refer their measurements to axes (x,y,z) that rotate with angular speed Ω‎ about the z axis. That is, they employ

(1.24)
x=xcosφ+ysinφ,y=xsinφ+ycosφ

and z=z, where φ=Ωt. Thus the orthonormal tetrad frame λ˜μαˆ of such static noninertial observers is given in (ct,x,y,z) coordinates by

(1.25)
λ˜μ0ˆ=(1,0,0,0),

(1.26)
λ˜μ1ˆ=(0,cosφ,sinφ,0),

(1.27)
λ˜μ2ˆ=(0,sinφ,cosφ,0),

(1.28)
λ˜μ3ˆ=(0,0,0,1),

so that at t=0, λ˜μαˆ coincides with the tetrad frame of the fundamental static inertial observers.

Let us next imagine an observer O moving uniformly for t<0 on a plane parallel to the (x,y) plane with x=r, y=vt and z=z0, where r and z0 are constant lengths and v=rΩ. At t=0, O is forced to rotate uniformly on a circle of radius r about the z axis on the plane that is at fixed z=z0 and is parallel to the (x,y) plane as in Fig. 1.1.

Introduction

Fig. 1.1 Schematic plot depicting the path of accelerated observer O that moves on the plane z=z0 with uniform speed v=rΩ along a straight line parallel to the y axis for t<0 but for t0 undergoes uniform rotation with angular speed Ω‎ about the z axis such that φ=Ωt.

For any instant of time t0, the natural orthonormal tetrad frame λμαˆ of O, adapted to the rotating system, can be simply obtained from that of the corresponding static observer O˜ by a pure boost with speed v=rΩ along the tangential direction to the circular trajectory (see Fig. 1.1); that is,

(p.7)

(1.29)
λμ0ˆ=γλ˜μ0ˆ+βλ˜μ2ˆ,

(1.30)
λμ1ˆ=λ˜μ1ˆ,

(1.31)
λμ2ˆ=γλ˜μ2ˆ+βλ˜μ0ˆ,

(1.32)
λμ3ˆ=λ˜μ3ˆ,

where β=v/c and γ=1/1β2 is the corresponding Lorentz factor. Thus, with respect to (ct,x,y,z) coordinates, λμαˆ is given by

(1.33)
λμ0ˆ=γ(1,βsinφ,βcosφ,0),

(1.34)
λμ1ˆ=(0,cosφ,sinφ,0),

(1.35)
λμ2ˆ=γ(β,sinφ,cosφ,0),

(1.36)
λμ3ˆ=(0,0,0,1).

This is the tetrad frame of the uniformly rotating observer O for t0. At each instant t0 along its circular trajectory in the orbital plane that is parallel to the (x,y) plane, (p.8) the spatial frame of O, λμiˆ, for i=1,2,3, consists of the radial, tangential and normal directions, respectively, with respect to the orbital plane.

The uniform speed of observer O is given by rΩ<c and its proper time can be written as τ=ct/γ if we assume that τ=0 at t=0; therefore, φ=γΩτ/c. Moreover, the observer’s acceleration, which was turned on at t=τ=0, can be turned off after a finite period of time. Such observers exist for 0<r<c/Ω, the boundary of this open cylindrical region in Minkowski spacetime is the light cylinder of radius L=c/Ω. The light cylinder is a timelike hypersurface; therefore, observers inside this cylinder can in principle communicate with the outside world without any difficulty. In the r=0 limit, eqns (1.33)–(1.36) with β=0 and γ=1 reduce to the tetrad frame of the noninertial observer O˜ that is at rest along the axis of rotation at z=z0.

We have thus far discussed the measurements of inertial observers. We are also interested in the measurements of accelerated observers. What do accelerated observers measure? What are the laws of physics according to accelerated observers? What is the generalization of Lorentz invariance that applies to accelerated observers? We now turn to a discussion of these issues.

1.1.3 Nonexistence of ideal inertial observers

The special theory of relativity is about the standard relativistic physics of Minkowski spacetime, where gravity has been turned off. Physical phenomena in each global inertial frame of reference involve ideal inertial observers as well as accelerated observers. Indeed, all actual observers are accelerated; that is, inertial observers, though of deep theoretical significance, do not in fact exist. There is a basic dichotomy here involving theory and experiment that is noteworthy: The basic laws of non-gravitational physics have all been formulated with respect to ideal inertial observers, yet the experimental basis of these laws—namely, the foundation of physical science—has been established via actual observers that are all accelerated. To set the foundation of physical science on a firm basis, a connection must be established between inertial and accelerated observers. Simply stated, the fundamental microphysical laws, such as the principles of quantum mechanics, have been formulated for nonexistent ideal inertial observers, while all actual observers are accelerated. The resolution of this dichotomy requires an a priori axiom that relates inertial and accelerated observers. The observational consequences of such an axiom should then be compared with experimental results.

Ideal inertial observers are supposed to move on straight lines with constant speeds from minus infinity to plus infinity in a global inertial frame and carry constant local reference frames. It is important to note that these theoretical assumptions regarding ideal inertial observers cannot be directly verified by experiment. For instance, distant past and future states of the universe are not directly accessible to experimentation. Furthermore, repeated observational attempts to determine that an object is indeed at rest or moves uniformly on a rectilinear path will produce disturbances that cause deviations from the state of rest or uniform rectilinear motion. The ideal inertial observers are thus hypothetical and have been introduced to embody the principle of inertia perfectly. Real observers in this global inertial frame are all accelerated and we need to determine what accelerated observers actually measure. In this treatment, (p.9) observers can be sentient beings or measuring devices. In either case, observers are classical macrophysical systems that are extended in space. Any real measuring device is subject to various limitations; for example, it may not operate properly under certain conditions. Moreover, an accelerated device is under the influence of various internal inertial effects that could, over time, affect its constitution and mode of operation. In practice, all such issues require careful consideration; however, for the purposes of this theoretical discussion, we generally follow the standard practice in the theory of relativity and represent an observer by a single timelike world line for the sake of simplicity. This is not considered to be a fundamental limitation; rather, it helps simplify the analysis. In fact, this notion of an elementary observer can then be extended to a reference system by considering a congruence of elementary observers that occupy a finite spacetime domain in a global inertial frame in Minkowski spacetime. That this construction is indeed possible has been demonstrated in various ways by explicit examples for simple accelerated systems (see Mashhoon 2008 and the references cited therein). A general method based on fiber bundles for the construction of such reference systems involving nonintegrable anholonomic observers has been discussed by Auchmann and Kurz (2014).

By employing pointlike observers in our theoretical treatment, we avoid the problem of determination of the integrated influence of inertial effects on measuring devices that are employed during the measurement process. All observers under consideration are thus essentially ideal pointlike systems subject to the laws of classical (i.e. nonquantum) physics. We can therefore concentrate on the theoretical distinction between pointlike inertial and accelerated observers.

Observational data, collected over time by actual observers that are all more or less accelerated, have helped establish microphysical laws and have indicated that Lorentz invariance is a fundamental symmetry of nature. Therefore, a connection must exist between inertial and accelerated observers. What is this connection?

1.2 Hypothesis of Locality

To extend Lorentz invariance to accelerated observers in Minkowski spacetime it is necessary to relate accelerated observers to inertial observers. The standard theory of relativity is based on the postulate that an accelerated observer is pointwise equivalent to an otherwise identical momentarily comoving inertial observer. The latter follows the straight world line that is instantaneously tangent to the world line of the accelerated observer. The locality postulate in effect replaces the world line of the accelerated observer at each instant by its tangent line at that event. Geometrically, the tangent line is the first Frenet approximation to the curve. The Frenet approach involves the mathematics of turning and twisting of a curve in space (O’Neill 1966). A discussion of the Frenet–Serret method of moving frames for world lines is contained in Synge (1971). The world line of the accelerated observer is the envelope of the set of straight tangent world lines; therefore, the accelerated observer may be replaced in effect by an infinite sequence of hypothetical momentarily comoving inertial observers. Thus the association between actual accelerated observers and ideal inertial observers is purely local, since an accelerated observer is pointwise inertial according to the standard theory of relativity.

(p.10) The hypothesis of locality originates from the Newtonian mechanics of point particles, where the state of a point particle is determined at each instant of time t by its position x and velocity v. The arbitrary point particle and the corresponding hypothetical momentarily comoving inertial particle of the same mass m share the same state (x,v) and are thus physically equivalent. The motion of the point particle of mass m under an external force f(t,x,v) in the background global inertial frame is given by

(1.37)
dxdt=v,dvdt=1mf(t,x,v).

The state of the particle (x,v) at time t uniquely determines the motion for all time. Moreover, if f is turned off at any time t, the motion continues uniformly on a straight line tangent to the path at t. The inertial tendency of the particle is thus continually interrupted by the presence of the external force, which changes the state of the particle. This is the physical explanation for the fact that the accelerated path of the particle under the external force is the envelope of the straight tangent lines. It follows that the postulate of locality is automatically satisfied in the Newtonian mechanics of point particles, as it is ingrained in the Newtonian laws of motion. Hence no new physical assumption is needed to deal with accelerated systems in Newtonian physics. The hypothesis of locality should hold equally well in the relativistic mechanics of classical point particles (Minkowski 1952). Moreover, it is clear that the hypothesis of locality would be exactly valid if all physical phenomena in Minkowski spacetime could be reduced to pointlike coincidences of classical point particles and rays of radiation (Einstein 1950). The hypothesis of locality is schematically illustrated in Fig. 1.2.

Introduction

Fig. 1.2 Schematic illustration of the locality postulate that an accelerated observer is pointwise inertial. The accelerated observer at each instant of time measures what an otherwise identical hypothetical momentarily comoving inertial observer would measure at that instant. They share the same state (x,v). The accelerated observer thus passes through a continuous infinity of tangent inertial observers; indeed, its path is the envelope of the corresponding family of straight lines.

It is through the locality postulate that the consequences of Lorentz invariance can be verified by experiment within the framework of the special theory of relativity. That is, it follows from the hypothesis of locality that in a global inertial frame in Minkowski spacetime the measurements of an accelerated observer can be determined by applying Lorentz transformation point by point along its path. Consider, for instance, the measurement of time by an ideal clock following an accelerated path. At each instant of time t, the clock has velocity v(t) and is instantaneously inertial by the locality hypothesis and hence at rest in an inertial frame with coordinates xμ=(ct,x). An instantaneous Lorentz transformation from the background inertial frame to the momentary rest frame of the moving clock leads to the formula for time dilation, namely,

(1.38)
dt=1v2c21/2dt,

which can also be obtained from the invariance of the Minkowski spacetime interval (1.2) under Lorentz transformations. Here, v=|v| and only positive square roots are considered throughout. The accelerated clock passes through an infinite sequence of such momentarily inertial states; therefore, its local proper time τ/c is a sum of infinitesimal time intervals each of the form of eqn (1.38). Hence, the proper time of the clock is given by

(1.39)
τc=0t1v2(T)c21/2dT,

where τ=0 at t=0 by assumption.

(p.11) We note that τ/ct in eqn (1.39). In this connection, imagine two identical ideal clocks at rest at some point P in space in the background inertial frame and synchronized to register time t=0. One clock remains at rest at P, while the other moves about and eventually returns to P at time t, the proper time of the clock at rest. At t, the clock that was in motion registers proper time τ/c given by eqn (1.39), which is shorter than t. For a discussion of clock experiments using the Global Positioning System (GPS), see Ashby (2003).

The influence of acceleration on clock performance has been discussed by a number of authors: see Mainwaring and Stedman (1993), Dahia and Felix da Silva (2015) and the references cited therein. Possible deviations from locality tend to be rather small and well below the level of accuracy (109) of the recent experimental verification of time dilation by Botermann et al. (2014).

Consider next an example of the application of the locality postulate involving length measurement. Imagine two small identical blocks A and B at rest and a distance L apart along the x axis in a global inertial frame. At t=0, A and B are accelerated from rest and forced to move in exactly the same way along the positive x direction. (p.12) The speeds of A and B are the same by assumption for all t0; therefore, the distance between A and B as measured in the inertial frame is always L. At time t>0, blocks A and B have the same speed v(t) and are momentarily inertial by the locality postulate; hence, an instantaneous Lorentz transformation connects the background inertial frame to the instantaneous inertial frame in which both A and B are at rest. It follows from this Lorentz transformation that at time t

(1.40)
xBxA=γ(t)(xBxA),γ(t)=1v2(t)c21/2,

so that the distance between the two blocks as measured in their momentary rest frame is generally larger than L and is given by γ(t)L, where γ1 is the Lorentz factor corresponding to speed v(t). If, for instance, there is a taut string that is initially attached to A and B and these are then forced to undergo hyperbolic motion in exactly the same way, the distance between A and B will continue to increase monotonically in their momentary rest frames and the string should eventually break (Dewan and Beran 1959; Bell 1987). It is assumed here that the string is always in tension but exerts negligible force on A and B. For a detailed analysis of the problem of length measurement in accelerated systems, see Mashhoon (1990b) and Mashhoon and Muench (2002).

1.2.1 Physics of locality

The postulate of locality states that acceleration can be locally ignored. This means, in terms of realistic measurements, that the integrated influence of inertial effects over the length- and timescales characteristic of the measurement process can be neglected. Hence, the observer’s tetrad frame should in effect remain constant during the process of measurement.

In retrospect, the hypothesis of locality appears rather simple and natural. For instance, Maxwell’s (1880) considerations regarding optical phenomena in moving systems implicitly contained the hypothesis of locality. Its approximate validity was later assumed by Lorentz (1952) in his theory of electrons in order to ensure that an electron, conceived as a small ball of charge, would always be Lorentz contracted along its direction of motion; see Section 183 of Lorentz’s book (1952). It was clearly recognized by Lorentz that this is simply an approximation whose validity rests on the supposition that the electron velocity would change over a timescale that is much longer than the period of internal oscillations of the electron, see p. 216 of Section 183 of Lorentz (1952).

A similar assumption was simply adopted by Einstein for rods and clocks as a useful approximation, see the footnote on p. 60 of Einstein (1950). In the early days of relativity theory, the locality assumption was discussed in terms of the “clock hypothesis", as it led to the so-called twin paradox; in this connection, Sommerfeld’s notes on Minkowski’s 1908 paper are quite informative, see p. 94 of Minkowski (1952). Indeed, the hypothesis of locality underlies Einstein’s development of the theory of relativity. For instance, the locality assumption fits perfectly together with Einstein’s local principle of equivalence to ensure that every observer in a gravitational field is pointwise inertial. In fact, to preserve the operational significance of Einstein’s heuristic principle of equivalence—namely, the presumed local equivalence of an observer in (p.13) a gravitational field with an accelerated observer in Minkowski spacetime—it must be coupled with a statement regarding what accelerated observers actually measure. When coupled with the hypothesis of locality, Einstein’s principle of equivalence provides a physical basis for a field theory of gravitation that is consistent with (local) Lorentz invariance.

Following Einstein’s development of the general theory of relativity, Weyl discussed the physical basis for the hypothesis of locality, see pp. 176–177 of Weyl (1952). In particular, Weyl noted that the locality hypothesis was an adiabaticity assumption analogous to the one for sufficiently slow processes in thermodynamics and would therefore be expected to be a good approximation only up to some acceleration.

If the hypothesis of locality is an approximation, what is the exact result? For instance, if the locality postulate is valid at sufficiently low accelerations, what happens at high accelerations? Can one devise an approximation scheme in which the locality postulate would be the first approximation? It appears that following the great success of general relativity, relativistic physics was generally considered to be simply local (Robertson 1949). The investigation of the difference between actual accelerated observers and the hypothetical inertial observers and the related problem of the domain of validity of the locality postulate received little or no attention until about thirty years ago (Mashhoon 1986, 1988, 1990a, 1990b).

1.2.2 Standard measuring devices

Ideal measuring devices that are so robust as to be essentially unaffected by acceleration are called “standard". Thus a standard clock measures proper time along its world line (“clock hypothesis"). From a modern perspective, all ideal measuring devices that are pointwise inertial are standard. That is, an ideal measuring device is practically standard if we can suppose that over the length- and timescales characteristic of typical measurements, the net impact of the internal inertial effects over the operation of the device can be neglected. Though cognizant of the possible limitations of these ideas, we will adhere to the traditional approach to spacetime measurements in relativity theory and assume that all measuring devices are standard. In this way, we will concentrate on the intrinsic nonlocality of the measurement of phenomena associated with electromagnetic fields. That is, as explained in the next chapter, even when an accelerated observer employs only ideal standard devices for measurement purposes, there are intrinsically nonlocal measurements involving electromagnetic fields that extend over the past world line of the observer and hence go beyond the postulate of locality.

1.2.3 Acceleration tensor

Inertial observers are endowed with local reference frames; therefore, it follows from the hypothesis of locality that an accelerated observer in Minkowski spacetime carries an orthonormal tetrad frame λμαˆ(τ) such that at each instant of proper time τ‎, this frame coincides with the tetrad frame carried by the momentarily comoving inertial observer. Here, λμ0ˆ=dxμ/dτ is the observer’s 4-velocity vector uμ, which is the unit timelike vector that is tangent to the path, and λμiˆ, i=1,2,3, are three unit spacelike vectors that constitute the local spatial frame of the accelerated observer moving in (p.14) a background inertial frame with inertial coordinates xμ=(ct,x). The operational establishment of the local tetrad frame of the accelerated observer is ultimately based on the standard rods and clocks that the accelerated observer may use for local spacetime determinations. It follows from the method of moving frames that

(1.41)
dλμαˆdτ=Φαˆβˆ(τ)λμβˆ,

where Φαˆβˆ is the acceleration tensor of the observer. The orthonormality of the frame field implies that Φαˆβˆ=Φβˆαˆ. This invariant antisymmetric acceleration tensor can be decomposed into spacetime scalars as Φαˆβˆ(g˜,Ω˜) in close analogy with the standard decomposition of the electromagnetic field tensor Fμν(E,B) into electric (E) and magnetic (B) components, where F0i=Ei, Fij=ϵijkBk and ϵ123=1 in our convention. Thus the translational acceleration of the observer is given by the “electric" part of the acceleration tensor, Φ0ˆiˆ=g˜iˆ(τ), while the angular velocity of the rotation of the observer’s spatial frame with respect to a locally non-rotating (i.e. Fermi–Walker transported) frame is given by the “magnetic" part of the acceleration tensor, namely, Φiˆjˆ=ϵiˆjˆkˆΩ˜kˆ(τ).

To clarify the interpretation of Ω˜, imagine that the accelerated observer carries a non-rotating orthonormal frame nμαˆ(τ) along its world line as well. That is, nμ0ˆ=λμ0ˆ=uμ and the spatial frame is Fermi–Walker transported in accordance with eqn (1.21), namely,

(1.42)
dnμiˆdτ=(nνiˆaν)nμ0ˆ.

The spatial frame of the observer can be obtained from the non-rotating frame by a time-dependent rotation,

(1.43)
λμiˆ=Miˆjˆnμjˆ,

where M(τ) is an orthogonal matrix. The instantaneous angular velocity of the rotation is a pseudovector Ω˜(τ) with respect to the local frame of the observer given by

(1.44)
dMiˆjˆdτ=ϵiˆkˆlˆΩ˜lˆMkˆjˆ,

which is consistent with the orthogonality of M(τ). It follows from eqns (1.42)–(1.44) that the rotational motion of the spatial frame is properly contained in the definition of the acceleration tensor in eqn (1.41). That is, differentiating eqn (1.43) and evaluating dλμiˆ/dτ via eqns (1.42) and (1.44), we recover the same equation for the variation of the spatial frame that is contained in eqn (1.41).

It is possible to define a spacelike 4-vector of angular velocity Ω˜μ that is orthogonal to uμ via

(1.45)
Ω˜μ=λμkˆΩ˜kˆ

and then it is straightforward to show that the acceleration tensor Φμν,

(1.46)
Φμν=λμαˆλνβˆΦαˆβˆ,

can be expressed as

(1.47)
Φμν=aμuνaνuμ+ϵμνρσuρΩ˜σ,

where ϵ0123=1 in our convention.

(p.15) In Newtonian mechanics, the state of a pointlike observer is given at any instant of time by its position and velocity in space; however, the situation is clearly different for a pointlike observer in relativity theory. The state of such an observer is given by its spacetime position along a future-directed world line and its adapted orthonormal tetrad frame. The four coordinates of the event together with the six independent components of the frame (i.e. three boosts and three rotations or, stated otherwise, sixteen tetrad components subject to ten orthonormality conditions) render the state space of the elementary observer a ten-dimensional manifold. The tetrad frame moves along the timelike world line in accordance with eqn (1.41). In this connection, it is interesting to note that historically the Frenet–Serret method of moving frames for curves in space was later extended to surfaces in space by Darboux; however, the recognition of the full power of this method and its complete generalization was accomplished by E. Cartan.

It seems intuitively clear that an accelerated observer in a global inertial frame of reference may be considered practically inertial during an experiment if the observer’s acceleration is such that its motion is in effect uniform and rectilinear and its spatial frame is non-rotating for the duration of the physical process under study. Lorentz invariance can then be employed to predict the result of the experiment. More generally, let λ/c be the intrinsic timescale for the process under consideration, and let L/c be the relevant acceleration timescale over which the tetrad frame of the observer changes appreciably; then, the condition for the validity of the hypothesis of locality is

(1.48)
λL.

The time and length scales over which the state of an accelerated observer changes are given by invariants that can be constructed out of g˜(τ), Ω˜(τ) and the speed of light c. For instance, an observer may have a translational acceleration length 1/|g˜| and a rotational acceleration length 1/|Ω˜|. For observers at rest on the surface of the Earth, c2/|g|1 light year and c/|Ω|28 astronomical units. These astronomical lengths are very large compared to laboratory dimensions of interest on Earth; hence, the hypothesis of locality is ordinarily a rather good approximation. For instance, in an optics experiment in the laboratory involving λ105 cm, we have λ/L1020. This means that for most physical situations the acceleration of the observer can be neglected for the duration of the physical process under consideration, which explains why locality is so effective in practice.

1.2.4 Local geodesic coordinates for accelerated observers

Imagine an accelerated observer in a global inertial frame in Minkowski spacetime. The observer follows the reference world line xˉμ(τ), where τ‎ is its proper time; moreover, it carries along this path an orthonormal tetrad frame λμαˆ(τ), where λμ0ˆ=dxˉμ/dτ is its unit temporal axis and λμiˆ, i=1,2,3, constitute its local spatial frame. The observer’s acceleration tensor is given by Φαˆβˆ(g˜,Ω˜), where g˜(τ) and Ω˜(τ) are respectively the tetrad components of the reference observer’s translational acceleration of its world line and the rotational angular velocity of its spatial frame with respect to the local non-rotating (i.e. Fermi–Walker transported) frame. Let us now consider a geodesic (p.16) system of coordinates Xμˆ=(cT,Xiˆ) established in a certain spacetime domain in the neighborhood of the world line of the reference observer. Given any event τ‎ along the fiducial path xˉμ(τ), the straight spacelike geodesic lines orthogonal to the reference observer’s world line at τ‎ span a simultaneity hyperplane that is in fact the three-dimensional Euclidean space. An event on this hyperplane with inertial coordinates xμ in the background global frame will be assigned geodesic (“Fermi") coordinates Xμˆ. The proper distance away from the reference world line is given by the length of the unique spacelike geodesic on the simultaneity hyperplane connecting xˉμ(τ) with xμ, see Fig. 1.3. The coordinate transformation xX is given by

(1.49)
τ=X0ˆ,xμ=xˉμ(τ)+Xiˆλμiˆ(τ).

Introduction

Fig. 1.3 Schematic construction of a geodesic system of coordinates Xμ=(cT,X) about an accelerated observer. The geodesic (Fermi) coordinates of an event xμ in the background inertial frame are determined by the way in which xμ can be connected orthogonally to the observer’s path via a geodesic (i.e. straight) line. If such a connection occurs at xˉμ(τ), then cT=τ and xμxˉμ(τ) has components X along the axes of the spatial frame of the observer at τ‎. A second possibility is indicated in the plot at τ; however, a coordinate system must uniquely identify events. Therefore, to avoid such a possibility, the spatial extent of the geodesic coordinate system is in general limited to a sufficiently narrow world tube along the timelike world line of the observer.

It follows from eqns (1.41) and (1.49) that

(1.50)
dxμ=(Pλμ0ˆ+Qjˆλμjˆ)dX0ˆ+λμiˆdXiˆ,

where

(1.51)
P=1+g˜X,Qiˆ=(Ω˜×X)iˆ.

It is then simple to show that the Minkowski metric tensor ημνdxμdxν with respect (p.17) to the new geodesic coordinate system can be written as gμˆνˆdXμˆdXνˆ, where

(1.52)
g0ˆ0ˆ=P2+Q2,g0ˆiˆ=Qiˆ,giˆjˆ=δiˆjˆ.

If the acceleration tensor of the observer vanishes, the geodesic coordinate system reduces to a global inertial frame that covers Minkowski spacetime and gμˆνˆ=ημˆνˆ. The geodesic coordinates are thus quasi-inertial; therefore, for the sake of notational consistency, the use of hatted indices has been extended here to the quasi-inertial geodesic (Fermi) coordinate systems constructed on the basis of the tetrad frames of the reference observers. It is important to note that the reference observer permanently resides at the spatial origin of the geodesic coordinate system, namely, X=0.

It is interesting to illustrate the construction of geodesic coordinate systems about the world lines of uniformly accelerated observers of Section 1.1. Let us first consider the fiducial world line xˉμ(τ) given by eqn (1.22). It follows from eqn (1.49) that an event with coordinates (ct,x,y,z) in the background inertial frame has geodesic (Fermi) coordinates (cT,X,Y,Z) such that

(1.53)
ct=(Z+c2g0)sinhΘ0+g0Tcc2g0sinhΘ0,

(1.54)
z=(Z+c2g0)coshΘ0+g0Tcc2g0coshΘ0,

x=X and y=Y. Here, the only nonzero components of the acceleration tensor are given by Φ0ˆ3ˆ=Φ3ˆ0ˆ=g˜3ˆ=g0/c2.

Let us next consider observers O˜ that are all at rest in the background global inertial frame, but refer their measurements to uniformly rotating axes. In this case, we have ct=τ and

(1.55)
xˉμ(τ)=(τ,x0,y0,z0).

Then, it follows from eqn (1.49) and eqns (1.25)–(1.28) that T=t,

(1.56)
X=(xx0)cosΩt+(yy0)sinΩt,

(1.57)
Y=(xx0)sinΩt+(yy0)cosΩt

and Z=zz0. For such a static observer, Φ1ˆ2ˆ=Φ2ˆ1ˆ=Ω˜3ˆ=Ω/c are the only non-zero elements of the acceleration tensor. Let us note that the standard classical rotating coordinate system (Landau and Lifshitz 1971) is thus the geodesic (Fermi) system established in the neighborhood of a static noninertial observer O˜.

Finally, we consider the uniformly rotating observer O with β=rΩ/c,

(1.58)
xˉμ(τ)=γτ,rcosγΩτc,rsinγΩτc,z0

and tetrad frame λμαˆ(τ) given by eqns (1.33)–(1.36). The corresponding coordinate (p.18) transformation in this case is given by

(1.59)
t=γ(T+1cβY),

(1.60)
x=(X+r)cos(γΩT)γYsin(γΩT),

(1.61)
y=(X+r)sin(γΩT)+γYcos(γΩT)

and z=Z+z0. In this case, the only non-zero elements of the acceleration tensor Φαˆβˆ(g˜,Ω˜) are given by the centripetal acceleration g˜1ˆ=γ2βΩ/c and the rotational angular velocity Ω˜3ˆ=γ2Ω/c, so that g˜Ω˜=0. We note that the spatial frame of the uniformly rotating observer rotates with respect to a corresponding non-rotating (i.e. Fermi–Walker transported) frame nμαˆ at a rate of γΩ with respect to time t of the static inertial observers. The tetrad frame λμαˆ rotates relative to static inertial observers at a rate of Ω‎ per unit time t; therefore, the non-rotating frame nμαˆ rotates with respect to the static inertial observers at a rate of (1γ)Ω per unit time t. This corresponds to the Thomas precession frequency, which can be interpreted as being ultimately due to an overcompensation as a consequence of time dilation involving proper time of static inertial observers and the proper time of the rotating observer (Mashhoon and Obukhov 2014).

In general, the spacetime manifold can be covered by an overlapping set of admissible coordinate charts. The admissibility of a system of coordinates xμ=(ct,xi) is related to the possibility of making proper temporal and spatial measurements by the static observers associated with the coordinate system, namely those that remain at rest in space. Let the spacetime interval take the form ds2=gμνdxμdxν in the system of coordinates under consideration. Then, the 4-velocity of the observers at rest is given by uμ=dxμ/dτ=(1/g00)δ0μ, since the proper time of the static observer is given by ds2=dτ2=g00(cdt)2. Thus the temporal admissibility condition requires that g00<0. Furthermore, consider the measurement of the length of an infinitesimal measuring rod by the static observer. The local hypersurface of simultaneity in the immediate neighborhood of the world line of a static observer is orthogonal to its world line and is given by uμdxμ=0, so that cg00dt+g0idxi=0. From the fact that

(1.62)
ds2=1g00(cg00dt+g0idxi)2+γijdxidxj,

where

(1.63)
γij=gijg0ig0jg00,

we see that the measured length of the infinitesimal rod is given by d2=γijdxidxj, where (γij) is a symmetric matrix. The spatial admissibility condition is then the requirement that (γij) be a positive-definite matrix. It turns out that the temporal and spatial admissibility conditions imply that the principal minors of the metric tensor (gμν) must all be negative in our (,+,+,+) convention. That is, (gμν) must be a negative-definite matrix. For further discussion and extension of these ideas, see Bini, Chicone and Mashhoon (2012).

(p.19) Returning now to geodesic coordinates in Minkowski spacetime, it follows from a detailed investigation (Bini, Chicone and Mashhoon 2012) that the new geodesic coordinates are admissible in a spacetime neighborhood around xˉμ(τ) so long as g0ˆ0ˆ<0. Equation (1.52) implies that the boundary of the admissible region, g0ˆ0ˆ=0, is given by P2=Q2, which is a quadratic equation in the spatial coordinates and represents a surface. Such surfaces have been classified under the Euclidean group into seventeen standard forms called quadric surfaces (O’Neill 1966; Birkhoff and MacLane 1953). If the reference observer is only translationally accelerated, the quadric boundary surface degenerates to coincident planes, given in the case of the hyperbolic observer Oˆ by Z=c2/g0. Turning next to static noninertial observers O˜, the boundary hypersurface of the admissible region is the circular light cylinder of radius c/Ω. Finally, for the uniformly rotating observer O, we note that the quadric surface can be expressed as

(1.64)
(X+r)2+γ2Y2=c2Ω2,

which for all Z is an elliptic cylinder whose axis coincides with the Z axis. For any constant Z, eqn (1.64) represents an ellipse with semimajor axis c/Ω, semiminor axis c/(γΩ) and eccentricity v/c. In fact, this ellipse appears as a circle of radius c/Ω that is Lorentz–Fitzgerald contracted along the direction of motion of O. For Z=0, the center of the ellipse is at x=y=0 and z=z0, and the reference observer is at one of the foci of this ellipse; as vc, rc/Ω, the reference observer approaches the light cylinder and the area of the ellipse tends to zero.

A general discussion of the boundary of admissible geodesic coordinates about the world line of an arbitrary accelerated observer is contained in the next chapter. For a detailed discussion of the properties of the boundary hypersurface, see Mashhoon (2003a); moreover, a general discussion of the inertial effects in geodesic coordinates and further references are contained in Chicone and Mashhoon (2005).

The quasi-inertial geodesic coordinate system is by no means the only possible way to establish coordinates in order to identify events uniquely in the neighborhood of an accelerated observer. Another useful method is furnished, for example, by radar coordinates (Bini, Lusanna and Mashhoon 2005). It is indeed possible to show that for observers whose world lines are infinitesimally close to each other, the results of the radar approach are identical to those based on the geodesic coordinate system. All such systems eventually break down, however, due to the existence of acceleration lengths of the observer (Bini, Lusanna and Mashhoon 2005). On the other hand, Minkowski spacetime can always be adequately covered by an overlapping set of such local coordinate charts.

1.3 General Relativity and Locality

The hypothesis of locality provides the simplest possible way to extend relativity theory to noninertial observers in Minkowski spacetime. The next fundamental step involves the extension of relativity theory to observers in a gravitational field. This is achieved via Einstein’s principle of equivalence. According to this heuristic principle, an observer in a gravitational field is presumed to be locally equivalent to a certain accelerated observer in Minkowski spacetime. The physical basis for this idea (p.20) is the universality of the gravitational interaction in the framework of the Newtonian laws of motion as well as Newtonian gravitation. The universality of gravity in turn follows from the principle of equivalence of inertial and gravitational masses. The principle of equivalence, which has firm observational support, originally provided Einstein with the key to the relativistic theory of gravitation. That is, Einstein interpreted the experimentally well-established principle of equivalence to mean that there is an intimate connection between inertia and gravitation. This notion eventually led to Einstein’s own extremely local principle of equivalence.

Einstein’s principle of equivalence and the hypothesis of locality, taken together, imply that observers in a gravitational field are all locally inertial. That is, Einstein’s principle of equivalence postulates a pointwise correspondence between measurements of an observer in a gravitational field with an accelerated observer in Minkowski spacetime, while the latter observer is pointwise equivalent to an inertial observer by the hypothesis of locality; therefore, an observer in a gravitational field is pointwise inertial. Thus, at all regular events in a gravitational field observers can define local inertial frames that are then somehow connected with each other through the structure of spacetime. This circumstance suggests that the gravitational field must be inherent in the spacetime structure. The flat Minkowski spacetime has no structure capable of accommodating a gravitational field. The simplest possible way to connect all such local inertial frames of reference for observers in the presence of gravitation is through the pseudo-Riemannian (i.e. Lorentzian) geometry of curved spacetime, where the gravitational field is then identified with the curvature of the spacetime manifold. That is, a curved spacetime manifold is at each event locally (i.e. pointwise) flat. In this general relativistic (GR) framework, free test particles and null rays follow geodesics of the curved spacetime manifold. It remains to establish, within this framework, the gravitational field equation and the correspondence of the resulting relativistic theory with the Newtonian theory of gravitation.

In GR, the curved spacetime interval is given by

(1.65)
ds2=gαβdxαdxβ,

where the pseudo-Riemannian metric tensor gαβ of Lorentzian signature cannot be transformed to the Minkowski metric tensor by a global change of coordinates unless the Riemannian curvature tensor vanishes, which would indicate the absence of a gravitational field. In the presence of spacetime curvature, the ten independent components of gαβ correspond to the ten gravitational potentials in GR. The correspondence with Newtonian gravitation can be established when we formally let c. In this way, there is a certain correspondence between the metric tensor and Newtonian gravitational potential ϕN. Moreover, it follows from the correspondence between the geodesic equation and Newton’s second law of motion of a test particle under gravitation that the Levi-Civita connection (0Γμνσ) is similar to the Newtonian gravitational acceleration (iϕN). Finally, the Jacobi equation corresponds to the tidal equation of Newtonian physics; hence, the Riemann curvature tensor (0Rμνρσ) is similar to the Newtonian tidal matrix (ijϕN). To complete the structure of a proper field theory of gravitation, we need the analog of Poisson’s equation of Newtonian gravity, namely,

(1.66)
2ϕN=4πGρ,

(p.21) where G is Newton’s constant of gravitation and ρ‎ is the density of matter. We note that Poisson’s equation connects the matter content of space with the trace of the Newtonian tidal matrix. The simplest generalization of eqn (1.66) in the context of curved spacetime is Einstein’s gravitational field equation of GR, namely,

(1.67)
0Rμν12gμν0R=8πGc4Tμν.

Here Tμν is the energy–momentum tensor of matter, the Ricci tensor, 0Rμν=0Rαμαν, is the trace of the Riemann tensor and 0R=0Rαα is the scalar curvature. Thus the mass–energy content of spacetime is connected via eqn (1.67) to the appropriate trace of the Riemann tensor in the gravitational field equation of GR. In this way, GR is a field theory of gravitation in close analogy with Maxwell’s field theory of electromagnetism.

Einstein’s gravitational field equation describes how material sources (including non-gravitational fields) produce spacetime curvature, just as Maxwell’s electrodynamics describes how the electromagnetic field is generated by electric charges and their currents. That is, in a global inertial frame in Minkowski spacetime, the electromagnetic field equations are given by

(1.68)
[ρFμν]=0,νFμν=4πcJˉμ,

where Jˉμ is the total current 4-vector associated with electric charges. The antisymmetry of the electromagnetic field tensor, Fμν=Fνμ, immediately implies that in eqn (1.68) electric charge is conserved, namely, μJˉμ=0. Thus we would expect that the conservation law for the source of the gravitational field, namely,

(1.69)
0νTμν=0,

where 0 denotes covariant differentiation, be an immediate consequence of the gravitational field equation. This is indeed the case, since eqn (1.69) is a direct consequence of the reduced Bianchi identity. Furthermore, it follows from the field character of the gravitational interaction in Einstein’s theory that Tμν=0 does not necessarily mean that spacetime is flat; indeed, the Riemann curvature tensor could be non-zero in a Ricci-flat spacetime due to the existence of gravitational waves.

The Poisson equation of Newtonian gravitation is a consequence of the inverse square force law, which is ultimately based on astronomical observations in the Solar System that originally led to Kepler’s laws of planetary motion. Einstein’s gravitational field equation has generalized eqn (1.66) into a consistent relativistic framework that is in good agreement with present Solar System as well as binary pulsar data (Shapiro 1980; Stairs 2003). Nevertheless, on small laboratory scales, for instance, questions remain regarding the validity of the inverse square law of gravitation; at present, efforts continue on resolving such experimental problems (Adelberger et al. 2003; Hoyle et al. 2004; Adelberger et al. 2007; Kapner et al. 2007; Little and Little 2014). It will turn out later in this book that the nonlocal aspect of the gravitational interaction can lead to possible deviations from the inverse square law on galactic scales. The resulting (p.22) nonlocal extension of GR may then be used to resolve difficulties in astrophysics and cosmology, such as the problem of the rotation curves of spiral galaxies.

It is in general rather difficult to solve the Einstein field equation, since it contains a coupled system of second-order nonlinear partial differential equations for the gravitational potentials gμν(x). Many classes of exact solutions of these equations are known in cases involving certain symmetries; however, the general solution is unknown (Stephani et al. 2003).

The simplest generalization of the gravitational field equation of GR involves the addition of a cosmological constant 0Λ, namely,

(1.70)
0Rμν12gμν0R+0Λgμν=8πGc4Tμν.

However, many other generalizations of the field equation of GR are possible. The main purpose of this book is to present a nonlocal generalization of GR, where the gravitational field is local but satisfies a partial integro-differential field equation. In this way, gravitation becomes history-dependent. It seems that in this extension of GR, the nonlocal aspect of the gravitational interaction may simulate dark matter. That is, according to nonlocal GR, what appears as dark matter in astrophysics and cosmology may essentially be a manifestation of the nonlocality of the gravitational interaction.

Finally, it is important to point out that the Einstein field equation can be derived from an action principle. Indeed, eqn (1.70) follows from the variational principle of stationary action SGR, namely, δSGR=0, where

(1.71)
SGR=Lg+Lmc38πG0Λgd4x.

Here,

(1.72)
Lg=c316πGg0R

is the Lagrangian density of the gravitational field, Lm is the Lagrangian density of matter and non-gravitational fields and g=det(gμν).

1.4 Fundamental Observers

In the transition from the flat Minkowski spacetime of special relativity to the curved spacetime of general relativity, the global inertial frames are supplanted with local inertial frames. Thus in GR, observers no longer have access to a global system of parallel axes in the presence of gravitation.

It is a characteristic feature of the nonlocal generalization of GR developed in this book that the gravitational degrees of freedom are carried by the sixteen components of the tetrad frame field adapted to fundamental observers. These form a class of observers throughout spacetime whose tetrad frame field is rendered globally parallel by virtue of the introduction of the Weitzenböck connection in addition to the Levi-Civita connection. Thus, in this extension of GR framework, two distant vectors in spacetime are parallel if they have the same components relative to their respective (p.23) local fundamental tetrad frames (“teleparallelism"). Moreover, to find a solution of the nonlocal gravitational field equation is tantamount to the determination of the tetrad frame field of the fundamental observers throughout spacetime.

In nonlocal GR, the complete absence of the gravitational field implies that the fundamental observers reduce to those defined in Section 1.1., namely, inertial observers that are all at rest in a global inertial frame in Minkowski spacetime with local tetrad frames whose axes are aligned with the global Cartesian spacetime axes of the background inertial frame of reference.