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Einstein on Gravity and LightGravitomagnetism


In relativity theory, gravitomagnetic effects are inertial or gravitational field effects that might be expected when there is relative motion between bodies. Some of these effects are currently included within standard "core" physics, some aren't.

The name comes from an analogy with electromagnetism, where the motion of an "electric" charge produces "magnetic" side-effects. In gravitomagnetism, the "moving charge" is inertial-gravitational, and the "gravitomagnetic" effect can be considered as being due to a distortion of a body's surrounding inertial-gravitational field. These effects can be considered consequences of the finite speed of gravitational signals: if a body changes its location or velocity while its previous signals are in flight, the way that its signals are distributed though space should be affected. If we consider the inertia of a body to be partly or wholly determined by its interactions with its environment, then interfering with the way that it communicates with that environment might alter its apparent inertial properties. 

The name "gravitomagnetism" is slightly misleading, in that it doesn't have anything directly do do with electromagnetism, or with conventional magnetism. 

Standard effects


A forcibly-accelerated observer feels gee-forces that appear to them as evidence of an apparent gravitational field. Applying the general principle of relativity, the observer is entitled to blame the existence of this field on the relative acceleration of the rest of the universe. The observer can say: "When the background stars accelerate, I feel an associated gravitational field". The principle of relativity then requires that the effect be mutual ("democratic principle"), so, as a result of the general principle of relativity, if we forcibly accelerate a mass, the acceleration is smudged out into the surrounding region, and should cause light and particles near to the mass to undergo a certain amount of acceleration, too.

As the body resists the applied acceleration, it tries to grab onto its surrounding region of spacetime, and nearby bodies in that region get pulled along. The forced acceleration of a body effectively causes a gravitational field around that body, acting in the same direction as the acceleration. From the point of view of the "forcibly accelerated" observer, it is this same effect, applied to the relative acceleration of all the background stars around them, that produces the apparent gravitational field and the geeforces that they feel.


Similar arguments say that a body should be more gravitationally attractive in its equatorial plane if it is seen to be rotating, and that this attraction should be strongest towards the side of the body that moves away from us. The observer can say, "When I see the background stars to be circling around me, I feel associated gravitational fields pulling me outwards in the rotation plane, and around in the sense of the rotation". 

When we fire a rocket directly upwards from the Earth's equator, the centrifugal effect makes it easier for the rocket to escape the Earth's gravity, and from the point of view of the Earth observer, we can re-describe the effect by saying that the shell of stars and galaxies rotating around the Earth appear to create an outward-pointing centrifugal field within it (with the centrifugal forces usually felt by rotating or swept objects being considered to be the results of these fields). If we then apply topological arguments to turn this effect inside out, or apply the principle of mutuality to say that the background stars must feel the Earth to produce a similar effect, we find that rotation must make a body more gravitationally attractive.  We can make a parallel argument that this additional gravitational mass is a consequence of the energy bound up in the object's rotation w.r.t its environment, E=mc2

The second effect that we notice with our launched rocket is that, if the rocket proceeds in pretty much a straight line with respect to the background stars, then from the point of view of the Earth observer, the rocket is moving away from the Earth in an expanding spiral that makes it circle the Earth, East-to-West, once every every 24 hours. For the Earth observer, the launched rocket was originally traveling in a straight line, but had its path deflected into a spiral by the rotating shell of surrounding matter.

This tells us that a rotating shell of matter must generate inside itself a Coriolis field that pulls other objects around with it, and if we then again turn the description inside out, or apply the principle of mutuality, we conclude that the general principle of relativity predicts that rotating bodies should twist gravitational fieldlines, so that nearby objects and light are pulled around.

rotational gravitomagnetic effects

Both effects can be visualised as fieldline effects: the results of the finite speed of gravity plus relative rotation causing a body's gravitational fieldlines to twist around it when it rotates relative to its environment.

The "twist" causes the rotating object's gravitational centre of gravity to appear offset towards its receding side, and as a result, when we surround the body with observers and ask them to point towards where the body's gravitation appears to be centred, those indications no longer point to the same location, but to locations that are all offset in the rotation plane, away from a common centre. The idealisation of a rotating mass is no longer a point, but a ring, and this is part of why we say that rotating black holes under GR1915 are idealised as containing a ring-singularity at their centre, instead of a  point-singularity.

In current physics, this rotational dragging effect is referred to as frame-dragging, and the effect that this should have on orbiting gyroscopes is referred to as the Lens-Thirring effect. An orbiting experiment to test this for this effect ("Gravity Probe B"), was launched in 2004, with the (delayed) results now hoped for some time in 2008. 

Non-standard effects


We can extrapolate from rotational effects to argue that gravitomagnetic effects should also drag light and matter between bodies that are moving with a simple velocity. In the rotational case, the receding, redshifted side of a rotating mass should pull at us more strongly than the approaching, blueshifted side, and if we turn this association into a general rule, we'll expect receding redshifted bodies to pull more strongly than approaching blueshifted ones. This association is already obviously true when the shift is due to gravitation rather than motion, and we might even decide that it would be interesting to treat the conventional motion shifts that we see as the result of the body's associated gravitomagnetic field.

Although the expected phenomenology of "velocity-dependent" gravitomagnetic effects seems to correspond well to the available physical evidence, the status of this class of effect under current theory isn't totally clear. The phenomena that we'd associate with velocity-dependent gravitomagnetic effects seem to already be in place within general relativity for bodies with strong gravitational fields, and general relativity agrees that a moving gravitational filed carries energy and momentum, and can "bump" nearby objects. Moving gravitational masses can exchange momentum with nearby bodies, and for the effect to be mutual, bodies that would normally be considered to have insignificant gravity must also be able to cause the effect. Technically, there should be no distinction between the rules of physics for strong-gravity and weak-gravity bodies.  

In practice, current textbook theory makes a distinction between "gravitational" and "non-gravitational" physics, and assumes that there is a realm where the interactions of bodies can be modeled by assuming that these sorts of dragging effects don't occur. Current textbook relativity theory uses special relativity to model physics in flat spacetime, and says that general theories need to reduce to the physics of SR over small regions to be considered credible. 

The argument for dividing physics up in this way is that since small bodies have such negligible gravitational fields, any effects due to modifications of those fields must be even tinier.  The counter-argument is that the general principle tells us that inertial and gravitational mass are different aspects of the same underlying property, and the inertial mass of a small object is not insignificant in physics -- it's perhaps one of the most important factors in most calculations. Setting the inertial mass of bodies to zero destroys most inertial physics.  We also know that when we send signals between moving water molecules, the motion of those molecules causes an offset in the speed of light (Fizeau experiment), so the sort of dragging effect that we'd expect if our third gravitomagnetic effect was real has already been verified.

The velocity-dependent gravitomagnetism seems to offer an alternative way of creating and regulating local lightspeed constancy in situations involving relative motion. However, since the idea would mean that special relativity's geometry and derivations were wrong, most physicists tend to say that we know that this third effect doesn't happen, or that if it does happen it must be too weak to bother with, or that if it does happen to a significant degree with strong-gravity bodies( momentum exchange, slingshot effect), that this must be a special effect that only applies when gravitational fields  are strong. 

Since special relativity is founded on the geometrical assumption of flat spacetime, the third gravitomagnetic effect hasn't been integrated into Twentieth-Century relativity theory.

General expression

In its most general form, gravitomagnetism can be thought of as a simple "smudging" of a body's kinetic energy and momentum out into the surrounding region, as a field effect. Gravitomagnetism then allows passing bodies to exchange energy and momentum via the coupling of their external gravitomagnetic fields, without undergoing a direct collision.

However, we don't seem to be able to apply this most general concept of gravitomagnetism within relativistic field theories based on special relativity. A fully-fledged gravitomagnetic model would treat moving-body problems as exercises in curved space or curved spacetime, while special relativity treats them as problems involving only flat spacetime.

Quotes on Gravitomagnetism and Field Theory 

" I hold in fact,
1.    That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
2.    That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
3.    That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
4.    That in the physical world nothing else takes place but this variation subject (possibly) to the law of continuity. "  
W.K. Clifford, "On the Space-Theory of Matter" (1876)

" What is to be expected along the line of Mach's thought?

1.     The inertia of a body must increase when ponderable masses are piled up in its neighbourhood.
2.     A body must experience an accelerating force when neighboring masses are accelerated, and, in fact, the force must be in the same direction as that acceleration.
3.     A rotating hollow body must generate inside of itself a 'Coriolis field', which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well.

We shall now show that these three effects ... are actually present according to our theory. ... "
Albert Einstein, "The General Theory of Relativity", 1921 Princeton Lectures
translated and published in The Meaning of Relativity (1956)

" On the basis of the general theory of relativity, on the other hand, space as opposed to "what fills space", which is dependent on the co-ordinates, has no separate existence. ... If we imagine the gravitational field, i.e. the functions gik, to be removed, there does not remain a space of the type (1), but absolutely nothing, and also no "topological space". For the functions gik describe not only the field, but at the same time also the topological and metrical structural properties of the manifold.  ...  There is no such thing as an empty space, i.e. a space without field.  ...  Space-time does not claim existence on its own, but only as a structural quality of the field. "
Albert Einstein, "Relativity: the Special and the General Theory" (1955)

" … space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. "
Albert Einstein, "Relativity: the Special and the General Theory" (1955) (note to the 15th edition)

" The vision of Riemann, Clifford and Einstein, of a purely geometrical basis for physics, today has come to a higher state of development, and offers richer prospects -- and presents deeper problems -- than ever before. "
John Archibald Wheeler, Geometrodynamics (1962)

" Einstein's general theory of relativity provides a number of ways to generate non-Newtonian gravitational forces. ... The three outlined here were probably known by Einstein before he published his paper on the principle of general relativity in 1916. They were first specifically derived by Thirring in 1918, and since then have been contained in nearly every text on general relativity  ...

It has been pointed out that a rotating mass will exert a force similar to centrifugal force on a stationary test body; also, if the test body is moving at some constant velocity v, it will experience an additional force which is proportional to the cross product of the angular velocity of the rotating mass and the linear velocity of the test body. ... 

In employing Einstein's Theory to investigate the effect of a large accelerated mass on a small test body, it is found that the accelerated body drags the test body along with it. ... "
Robert L. Forward, "Guidelines to Antigravity" American Journal of Physics 31 166-170 (1962)

... the gravitational spin-spin force has the same form as the force between two dipoles in electromagnetism except that its sign is opposite, i.e., "north pole" attracts "north pole" in gravitational spin-spin interaction.
Robert Wald, "Gravitational Spin Interaction" Physical Review D 6 406-413 (1972)

" In other words, inside a rotating universe, the axes of gyroscopes rotate in step with the rotation ... they are tied to the directions of distant bodies in that universe. ... The existence of the dragging of inertial frames then guarantees that rotation must be defined relative to distant matter, not relative to some absolute space. ... "
Clifford M. Will, Was Einstein Right? (1986)
Chapter 11: "The Frontiers of Experimental Relativity"

... Intuitively intrinsic gravitomagnetism may be thought of as that phenomenon such that the spacetime geometry and curvature change due to currents of mass-energy relative to other matter.
Ignazio Ciufolini and John Archibald Wheeler Gravitation and Inertia (1995)
Section 6.11 "Gravitomagnetism, dragging of inertial frames, static geometry, and Lorentz invariance"

" As electric charge, going round and round in a circle, produces magnetism, so mass, going round and round in a circle, must produce a new kind of force, gravitomagnetism. ... The quickest way to an order of magnitude estimate of this Einstein-Lense-Thirring "frame dragging" is a line of reasoning invented by Mach which was very influential in guiding Einstein to his geometrodynamics. ... "
John Archibald Wheeler, A Journey into Gravity and Spacetime (1990, 1999)
Chapter 13: "A Farewell Look at Gravity"

" Looking at a "frozen instant" of an object moving through its environment, we still have a problem of interpretation: the photograph of the ball certainly could be interpreted as showing the ball moving, but it could also be interpreted as showing a ball immersed in a polarised gravitational field. Which interpretation is correct?
In a unified model, the two descriptions may be interchangeable. Is the redshift seen in a receding object due to its recession or to its gravitomagnetic dragging effect on light? As long as we can do the same calculations in each case and get the same answer, we really don't care. "

" If gravitomagnetic distortions are fundamental, a physical observer's perception of the apparent alignment of space and time axes is no longer just a matter of the mathematical projection of different coordinate systems onto a flat empty region: it becomes a more physical, visceral interaction between the observer and their environment. "
Eric Baird, Relativity in Curved Spacetime (2007)
section 9.13: "Zeno revisited: The 'impossibility' of motion" / "motion without motion"
section 9.14: "Worldlines and curvature"