universe. This idea, which has come to be calledMach’s Principle, was rather vague in Mach’s writings [10], and has triggered a large number of more precise formulations, with a wide range of mathematical and physical content [11]. The simplest and most straightforward version of Mach’s Principle states that, when there are no nearby, massive spinning bodies (such as the Earth, a neutron star, or a black hole), inertial axes are tied to the mean rotational motion of the matter that fills the distant universe.
Does general relativity satisfy this simple version of Mach’s Principle? The order-of-magnitude frame- dragging formula (5.2) suggests that it might. In order of magnitude, the universe’s Hubble distance (the size of our cosmological horizon) is equal to the Schwarzschild radius of all the matter within that distance, so Eq. (5.2) withR ∼2GM/c2andr∼ Rsuggests that (far from heavy spinning bodies, such as the Earth or Sun) inertial axes might rotate with the same angular velocity as the mean motion of our universe’s matter, ωdrag∼Ωuniverse—which could account for the observed fact that inertial axes are tied to distant stars.
Many mathematical models of our universe have been developed to explore whether this is so. Perhaps the simplest and most compelling was a 1966 study by Brill and Cohen [12], in which the universe’s matter is idealized as contained in a thin, spherical shell that rotates slowly with respect to inertial frames far outside it (at radial infinity). If the shell is arbitrarily close to its own Schwarzschild radius (analogous to the situation with our universe), the shell fully controls the inertial axes inside it: they turn in lock-step rotation with the shell. If the shell is very large compared to its Schwarzschild radius (by contrast with our universe), the standard of inertia at spatial infinity controls the interior inertial axes; they point toward ”stars” at rest at spatial infinity regardless of the shell’s rotation rate. As the shell is slowly shrunk from very large to its Schwarzschild radius, the inertial axes slowly transition from control by “infinity” to control by the shell.
This 1966 study added credence to the hope that General Relativity might incorporate Mach’s Principle in a very clean and compelling way. However, subsequent studies, using more realistic models of the universe, have revealed a more complicated and less satisfying picture [13, 14, 15, 16, 17, 18]. In Sec. 5.3 of this paper we present a simple and pedagogically illuminating variant of these types of studies.
5.2 Frame dragging: The space-drag and gravitomagnetic descrip-
to the distant stars (φ =constant), its time to travel around the fiber and return to the laser can be deduced by settingds2 =−c2dt2+r2sin2θ(dφ−ωdt)2 =0 along its path. (This is the only property of the general relativistic metric that we shall need: as in special relativity, so in general relativity, the intervalds2vanishes along photon trajectories.) Solving fordtand integratingφfrom 0 to 2π, we find for the clockwise travel time
∆t=2πrsinθ/(c−ωrsinθ). By contrast, if the photon travels counter-clockwise, its round-trip travel time is
∆t=2πrsinθ/(c+ωrsinθ).
To ensure that the travel times clockwise and counterclockwise are the same, we must set the laser into ro- tational motion with angular velocitydφ/dt=ω. A straightforward calculation based onds2=0 then reveals equal round-trip travel times. Correspondingly, the laser gyroscope identifiesdφ/dt = ωas a nonrotating motion, so far as inertia of photons is concerned.
It is convenient and fruitful to interpret this result as telling us that the Earth’s spin drags space into rotational motion relative to the distant stars (relative to inertial frames at radial “infinity”) with angular velocity dφ/dt=ω=(2G/c2)J/r3; and a laser gyroscope with its laser at rest with respect to this space sees clockwise and counter-clockwise light travel as taking the same round trip time. In other words, this laser gyroscope reveals to us (i.e. defines for us) the rotational state of space in the vicinity of the Earth.
One might object that Lorentz invariance (or, in general relativity, local Lorentz invariance) insists that the laws of physics cannot pick out any preferred state of motion, so it should not be possible or sensible to speak of space as having some preferred motion. This is certainly true oftranslationalmotion in special relativity and local translational motion in general relativity. However, it isnottrue of rotational motion in special relativity, or in axially symmetric situations like ours in general relativity. Rotational motion is very different from translational motion. Our laser gyroscope demonstrates this: it readily picks out a preferred
“rest frame” for space, so far as rotational motion is concerned.
Note that the rotational angular velocityω=(2G/c2)J/r3of space relative to the distant stars depends on radius. It is greater near the Earth than far away, and its associated linear velocity [which is the same as the function−~γin the Cartesian version (5.5) of the spacetime metric]
~vspace = −~γ = ωrsinθ ~eφ = 2G c2
1
r2 J~×~er (5.6)
is also greater near Earth than far away. This is similar to the velocity of the air in a tornado or the water in a whirlpool: greater near the center than far away. This analogy is more than heuristic. It can be made quantitative:
In the reference frame of any local observer who moves with the fluid, the velocity field of nearby fluid can be decomposed into three physically distinct parts: an isotropic expansion (volume change), a shear, and a local rotation (see textbooks on fluid mechanics, e.g. Chap. 12 of [19] and Sec. 6.4 of [20]). The angular velocity of the local rotation is equal to half the fluid’s vorticity, i.e., 12~∇ ×~v. A small, isotropic object (leaf, chip of wood, etc) placed in the fluid is dragged by the fluid into rotation with precisely this angular velocity.
Similarly, it is reasonable to expect that above the spinning Earth, any object whose orientation is inertially controlled (e.g. a gyroscope or a small nonspinning rock) will be dragged by the flow of space into rotation with a “frame-dragging” angular velocity given by
~ωdrag= 1 2
∇ ×~ ~vspace. (5.7)
By inserting expression (5.6) for~vspace, we obtain precisely the frame-dragging angular velocity predicted by general relativity, Eq. (5.1).
5.2.2 Gravitomagnetic description
When gravity is weak (linearized) and the velocities of all gravitating matter and test matter are small com- pared to the speed of light in the gravitating matter’s mean rest frame (slow motion), then the equations of general relativity can be rewritten in a language and notation that resembles Maxwell’s electromagnetic theory. For full details see, e.g., Refs. [21, 4, 6] and papers cited therein.
This Maxwell-like formulation of slow-motion, linearized general relativity entails writing the slow- motion geodesic equation for a test particle dvj/dt = −Γj00 −2Γj0k(vk/c) [with Γj0k = 12(γj,k−γk,j)] in Lorentz force notation:1
d~v dt = m
m ~g+~v c×H~
!
=~g+~v
c×H~ , where H~ =~∇ ×~γ . (5.8) Evidently, the Newtonian gravitational acceleration~gis the analog of the electric field (it is sometimes called the gravitoelectric field), and H~ is the analog of the magnetic field (and so is called the gravitomagnetic field). In electromagnetism the right hand side is multipled by the particle’s charge-to-mass ratioe/m. In the gravitational case the analog of charge is mass, so the right hand side is multiplied bym/m=1.
Just as the ordinary magnetic field is the curl of a vector potential A~ which (as dictated by Maxwell’s equations) is generated by the motion of electric charge, i.e. by the electric current density~j,
B~=∇ ×~ A~, ∇2A~=−4π~j, (5.9)
so the gravitomagnetic field is the curl of a vector potential~γwhich (as dictated by Einstein’s equations) is generated by the motion of mass, i.e. by the mass current densityρ~v(whereρis mass density and~vis the mass’s velocity):
H~ =∇ ×~ ~γ , ∇2~γ= +16πρ~v. (5.10)
The sign difference in the magnetic (5.9) and gravitomagnetic (5.10) source terms is due to the gravitational
1The geodesic equation can be written asduα/dτ=−Γαβγuβuγ. In slow-motion, linearized gravity, the 4-velocity takes its Newtonian limitu0 =1,uj =uj =vj, the proper timeτgoes to Minkowski timet, andΓjk0 'Γjk0. Therefore to first order in~v, the geodesic equation becomesdvj/dt=−Γj00−2Γj0k(vk/c).
force between like particles being attractive by contrast with the repulsive electromagnetic force; the factor 4 can be traced to the “spin-2” nature of weak gravity, viewed as a canonical field theory, by contrast with the spin-1 nature of electromagnetism.
Just as a spinning, charged sphere generates a dipolar magnetic field via Eq. (5.9), so the spinning Earth generates a dipolar magnetic field via Eq. (5.10):
~γ=−2G c2
1 r2
J~×~er, H~ =−2G c
−J~+3(J~·~er)~er
r3
, (5.11)
whereJ~is the spin angular momentum (i.e., the current dipole moment) and is defined asJ~=R
~r0×ρ~v d3~r0. Since the gravitomagnetic force on a mass element inside a gyroscope has the standard Lorentz-force form (~v/c)×H, the spin of a gyroscope precesses around a gravitomagnetic field in the same manner as a spinning~ particle with a magnetic dipole moment precesses around an ordinary magnetic field [22]. Translating the magnetic precession equation to the gravitomagnetic case,2we obtain for the gyroscopic precessional angular velocity
~ωdrag=−1
2cH~ = G c2
−J~+3(J~·~er)~er
r3
, (5.12)
which agrees with the standard general relativistic prediction (5.1) and with the fluid like dragging-of-space prediction (5.7).
5.2.3 Relation of space-drag and gravitomagnetic descriptions of frame dragging
The relation between the space-drag and gravitomagnetic descriptions of frame dragging becomes clear when one ties them both to relativity’s curved-spacetime description. The tie is through the off-diagonal compo- nentsg0j =γjof the frame-dragging metric (5.5): The velocity of space dragging is~vspace = −~γ, and the gravitomagnetic vector potential is~γ; so~vspaceand−~γare the same quantities but viewed in two very differ- ent physical pictures. Correspondingly, the vorticity of the flow of space~∇ ×~vspaceis the same as the negative of the gravitomagnetic field−H~ = −∇ ×~ ~γ. And finally, the physical fact that a gyroscope is dragged by the flow of space into precession with angular velocity~ωdrag = 12∇ ×~ ~vspaceis completely equivalent to the precession of the gyroscope around the gravitomagnetic field with angular velocity~ωdrag=−12H.~
2A magnetic dipole~µis defined as~µ≡12R
~r0×~j d3~r0, where~jis the electric current density (analogous toρ~v). In a magnetic field B, a small particle with magnetic dipole moment~ ~µfeels a torque~µ×B~(which follows from the Lorentz force law). Similarly in the gravitational case, the Lorentz-force-like geodesic equation (5.8) dictates that a gyroscope with spinS~feels a torque given by12S~×H/c.~ The gyroscope’s spin angular momentum is changed by this torque: dS~/dt = 12S~×H/c; i.e.~ S~rotates with the “gravitomagnetic”
angular velocity~ωGM=−H/2c.~