The Earth does not drag spacetime around with it like a whirlpool
Gravitoelectromagnetism, frame dragging myths and misconceptions
Einstein first mentioned a concept he called “dragging” in a 1913 letter to Mach, two years before he published his final theory of General Relativity that remains with us to this day. In it Einstein was referring to the plane of a Foucault pendulum being “dragged around”. In Newtonian physics, a Foucault pendulum, a pendulum with a long rope and heavy weight such as this one
maintains its plane of swing because of inertia while the Earth rotates. This effect gives the impression that the plane of swing rotates every 24 hours when in fact it is the Earth under it that rotates. General relativity, however, says that this inertial effect is not perfect and that the plane of inertia of the pendulum will itself be dragged by the rotation of the Earth because spacetime itself is turning.
Since that time, until the present day, numerous different dragging effects have been categorized and most observed. These include the dragging of angular velocity of inertial frames and dragging of the compass of inertia, called the Lense Thirring effect where the axis of a spinning gyroscope precesses when near a spinning massive body.
Other dragging effects apply to observers with Zero Angular Momentum which in GR paralance we dub ZAMOs. We can observe these to have some angular velocity with respect to distant stars. Meanwhile, objects with zero angular velocity can have some angular momentum even though they are not rotating.
Yet another effect is tidal dragging. This is where you have a gyroscope that precesses with respect to a coordinate frame defined by a set of gyroscopes at right angles to one another that are very close to it.
Physicists try to intuitively understand and explain frame dragging using two analogies. The first is electromagnetism and the second is based on fluids. While the first analogy is well supported even in strong fields like near black holes, the second, which is more common in the popular science press, is problematic.
The idea is that you can imagine spacetime as being like a fluid, and, as a body such as the Earth rotates, it drags that fluid around it and generates many of the effects we observe. While qualitatively true for some effects like those on ZAMOs, it is misleading and creates many misconceptions about how frame-dragging works.
The term frame “dragging” is unfortunately a misnomer that comes from this fluid analogy. When the Earth rotates, it does not “drag” spacetime about with it. If, for example, we were to build a ring around the Sun, as in Larry Niven’s classic novel Ringworld, the Sun would not drag the ring around with it at all, as you might expect if the fluid analogy were valid. If the ring were static with respect to the distant stars, it would stay static. More on this later.
We refer to frame-dragging as gravitoelectromagnetism since the first analogy is sound. This is not the same as, for example, charged bodies like Kerr-Newman black holes where you have electromagnetism coupled to gravity in a strong field which, clearly, might also be dubbed gravitoelectromagnetism. In the context of trying to understand how charged gravitating bodies behave dynamically, perhaps we should call these situations magnetogravitodynamics in analogy to magnetohydrodynamics.
Gravitoelectromagnetism works like this:
Gravitoelectric fields are governed by the spatial variations in the forward motion of time itself where this spatially varying “time field” is analogous to the electric potential. This is responsible for the Newtonian force we know and love.
Gravitomagnetic fields are governed by spatial variations in spatial shifts as time moves forward where the spatially varying field of shifts in the three spatial coordinates is analogous to the magnetic vector potential. This field is very weak in the Solar System.
A spatial shift can be thought of as how, when we observe an object at rest, as it moves forward in time it also moves along one of our spatial coordinate axes. This means that, while the object is at rest, it is propelled in space. The most familiar example of this is a warp drive. A warp bubble creates a space where an object is at rest inside, yet the entire bubble shifts in space as it moves in time, enabling the object inside to propagate at any velocity, even faster than light. Spatial shifts in dragging are much slower than light speed of course, but it is the same principle.
As in electromagnetism, the gravitoelectric field and the gravitomagnetic field couple together to create a compass of inertia dragging effects like the Lense-Thirring effect. This is a first order effect because the operational fields are first derivatives of the potentials.
There are two other sources of effects however. There are zeroth order. These even occur when the shifting vector is constant and there is no gravitoelectric field at all. The ZAMO effects, for example, where we observe angular velocity when no angular momentum is present, don’t require any variation in the field (in a spherical coordinate system). Another called the Sagnac effect is also in this category.
The Sagnac effect is one way to tell, even in an empty space, that you are rotating since rotation, unlike translation, is absolute. To produce this effect you fire two beams of light in opposite directions around a closed loop using, for example, a length of optical fiber cable. You then measure the Sagnac coordinate time delay if one light beam completes the loop faster than the other. The only way that this delay is zero, meaning the light beams finish at the same time, is if the observer has no angular momentum (ZAMO). Thus, the Sagnac delay tells the observers that they have angular momentum even if they think they are at rest.
There are also second order effects. These are “tidal”, meaning that they are caused by differential dragging. Unlike a first order effect, which is caused by variations in shifting and time, these are caused by variations in those variations. These are tidal because they are caused by spatial variations in the gravitomagnetic and gravitoelectric forces (which are first derivatives of a potential) rather than forces themselves.
This means that the compass of inertia is differential dragged when this is non-uniform. To measure this, we use four gyroscopes. Three of them are at right angles to one another in order to create a coordinate system. As they propagate about a massive spinning body, such as being in orbit about the Earth, these axes will precess, meaning that their inertial plane will appear to change. The fourth gyroscope, by the first order effect, would precess by the same amount and hence remain pointing in the same direction relative to that coordinate system. But tidal dragging, the second order effect, says that this is not so, and it will in fact precess relative to the local coordinate system!
These are all the possible effects, and they have all been observed to occur.
Going back to our ring world, we can now talk about why dragging is such a poor analogy. On such a world, you wouldn’t see any first or second order effects. The only one you would see on the ring world is the Sagnac effect, so if you were to run optical fiber around it, the beams of light would take different amounts of time going with the orbit of the planets versus against it. Thus, mental pictures of whirpools of spacetime about the Sun, Earth, black holes, or any such body are faulty and misleading. The effects from gravitoelectromagnetism only affect bodies that are moving with respect to the chosen reference frame and they are at right angles to the velocity of those bodies, exactly the opposite of dragging in a viscous fluid.
Thus, it is better to say that the rotation of the Sun induces a gravitoelectromagnetic field.
If we were to build a smaller ring, close to a Kerr black hole (a black hole with non-zero angular momentum), also at rest with respect to the distant stars, the same situation applies as to our ring world about the Sun, but the effects would be much larger. We could observe gravitoelectromagnetic effects only when we move objects around the ring. If we throw a ball, the gravitomagnetic field would suddenly go into operation. Since the black hole’s gravitomagnetic field points along its “north pole” much like an ordinary magnetic field, if we throw a ball towards the black hole, it would immediate be deflected in the direction of the black hole’s rotation. If, however, we threw the ball in the opposite direction, away from the black hole, it would be deflected in the opposite direction, opposing the black hole’s rotation. Would such an effect be seen if the spacetime were actually dragged around it like a whirlpool?
Likewise, if an object is launched into orbit about the black hole, the gravitoelectromagnetic field will not accelerate the object along in the direction of rotation. Rather, the force would be at a right angle to the orbit. If the orbit is with the rotation of the black hole, the force is repulsive, decreasing the overall Newtonian (gravitoelectric) field. If the orbit is opposing the rotation of the black hole, the force is attractive, increasing the Newtonian force.
Likewise, if we had a rotating spacestation ring around a non-rotating black hole, the station will not “drag” the black hole into rotating. Rather, the spacestation would induce on the black hole a non-zero angular momentum, but the black hole would still not be rotating.
One of the weirdest effects that is completely opposed to the frame dragging analogy is that, because of the possibility for gravitomagnetism to be repulsive, you can have it exactly counterbalance the gravitational attraction so that a test particle will remain stationary above the black hole as if it isn’t even there.
Given all this evidence against it, frame dragging is as poor an analogy for what is going on with gyroscopic precession experiments as “rubber sheet'“ analogies are for how general relativity explains gravity itself. Scientists and science writers can do better.