Gravitoelectromagnetism

Gravitoelectromagnetism, or GEM, is a way to compare the rules of electricity and magnetism with the rules of gravity. It shows how the math for electric and magnetic fields looks very similar to the math for gravity when we use a special version of Einstein's ideas about relativity.

Diagram regarding the confirmation of gravitomagnetism by Gravity Probe B

In simple terms, just like moving electric charges can create magnetic fields, moving masses can create effects like magnetic fields. These are called gravitomagnetic effects. This idea was first shared in 1893 by a scientist named Oliver Heaviside, before Einstein's big theories about relativity were even made.

GEM helps scientists understand gravity in new ways by using ideas from electromagnetism. It works best when we look at things far away from big masses and when those masses are moving slowly. This comparison makes it easier to study and predict some gravitational behaviors.

Background

This idea looks at gravity in a special way, like how we understand electric and magnetic fields. It shows that around a moving or spinning mass, there are two kinds of fields: one like the electric field, called the gravitoelectric field, and another like the magnetic field, called the gravitomagnetic field.

One important effect of the gravitomagnetic field is that objects moving near a large, spinning mass will act a little differently than we’d expect from regular gravity alone. Scientists have seen signs of these effects in powerful streams of energy from space, called relativistic jets, which come from very large and bright objects like quasars and active galactic nuclei.

The Gravity Probe B satellite tested these ideas and found evidence that matches what general relativity predicts about how Earth’s spin affects space around it.

Equations

According to general relativity, a spinning object creates a gravitational field that can be described with equations that look like those in classical electromagnetism. By starting with the main idea of general relativity, the Einstein field equation, and thinking about weaker gravity or space that isn’t curved too much, we can find rules similar to Maxwell's equations. These are called "GEM equations".

In these equations:

E_g is the gravitoelectric field (regular gravitational field), measured in meters per second squared (m⋅s⁻²)
E is the electric field, measured in kilograms meters per second cubed per ampere (kg⋅m⋅s⁻³⋅A⁻¹)
B_g is the gravitomagnetic field, measured in per second (s⁻¹)
B is the magnetic field, measured in kilograms per second squared per ampere (kg⋅s⁻²⋅A⁻¹)
ρ_g is mass density, measured in kilograms per cubic meter (kg⋅m⁻³)
ρ is charge density, measured in ampere seconds per cubic meter (A⋅s⋅m⁻³)
J_g is mass current density or mass flux, measured in kilograms per square meter per second (kg⋅m⁻²⋅s⁻¹)
J is electric current density, measured in amperes per square meter (A⋅m⁻²)
G is the gravitational constant, measured in cubic meters per kilogram per second squared (m³⋅kg⁻¹⋅s⁻²)
ε₀ is the vacuum permittivity, measured in kilogram inverse times cubic meter times second to the fourth times ampere squared (kg⁻¹⋅m⁻³⋅s⁴⋅A²)
c is both the speed of propagation of gravity and the speed of light, measured in meters per second (m⋅s⁻¹).

Potentials

We can describe the gravitoelectric and gravitomagnetic fields using special math tools called potentials. These help us solve the equations more easily.

Lorentz force

For a small object with mass, the total push or pull it feels in a GEM field can be described using an idea similar to one used in electricity and magnetism.

Poynting vector

The GEM version of a special energy flow idea used in electromagnetism looks similar to the one used in electricity and magnetism.

Scaling of fields

Scientists don’t all agree on how to compare the gravitoelectric and gravitomagnetic fields, which makes it a bit tricky. This is because gravity comes from something more complex than the source of electric fields.

Comparison of GEM and Maxwell's equations
Law	GEM equations	Maxwell's equations
Gauss's law	∇ ⋅ E g = − 4 π G ρ g {\displaystyle \nabla \cdot \mathbf {E} _{\text{g}}=-4\pi G\rho _{\text{g}}\ }	∇ ⋅ E = ρ ε 0 {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}
Gauss's law for magnetism	∇ ⋅ B g = 0 {\displaystyle \nabla \cdot \mathbf {B} _{\text{g}}=0\ }	∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0\ }
Faraday's law of induction	∇ × E g + ∂ B g ∂ t = 0 {\displaystyle \nabla \times \mathbf {E} _{\text{g}}+{\frac {\partial \mathbf {B} _{\text{g}}}{\partial t}}=0}	∇ × E + ∂ B ∂ t = 0 {\displaystyle \nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0}
Ampère–Maxwell law	∇ × B g − 1 c 2 ∂ E g ∂ t = − 4 π G c 2 J g {\displaystyle \nabla \times \mathbf {B} _{\text{g}}-{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} _{\text{g}}}{\partial t}}=-{\frac {4\pi G}{c^{2}}}\mathbf {J} _{\text{g}}}	∇ × B − 1 c 2 ∂ E ∂ t = 1 ε 0 c 2 J {\displaystyle \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}={\frac {1}{\varepsilon _{0}c^{2}}}\mathbf {J} }

Lorentz force equations in GEM and electromagnetism
GEM equation	EM equation
F g = m ( E g + v × 4 B g ) {\displaystyle \mathbf {F} _{\text{g}}=m\left(\mathbf {E} _{\text{g}}\ +\ \mathbf {v} \times 4\mathbf {B} _{\text{g}}\right)}	F = q ( E + v × B ) {\displaystyle \mathbf {F} =q\left(\mathbf {E} \ +\ \mathbf {v} \times \mathbf {B} \right)}

Poynting vector equations in GEM and electromagnetism
GEM equation	EM equation
S g = − c 2 4 π G E g × 4 B g {\displaystyle {\mathcal {S}}_{\text{g}}=-{\frac {c^{2}}{4\pi G}}\mathbf {E} _{\text{g}}\times 4\mathbf {B} _{\text{g}}}	S = c 2 ε 0 E × B {\displaystyle {\mathcal {S}}=c^{2}\varepsilon _{0}\mathbf {E} \times \mathbf {B} }

Higher-order effects

Some special effects in gravitomagnetism can act like forces between regular objects that push or pull each other. For example, if you spin two wheels on the same line, the force pulling them together can be stronger if they spin in opposite ways.

Gravitomagnetic ideas also suggest that a flexible or flowing round object that spins in a special way can pull matter through its center. In theory, this might help move things without feeling strong forces.

Imagine a round object that spins in two ways at once: around its middle and also like a spinning ring. This creates a twisty gravitational field around it. The forces pushing and pulling on this object can be tricky to understand because they happen at different places. Scientists have not yet modeled this complex behavior using curved space ideas.

Gravitomagnetic fields of astronomical objects

Rotating objects, like planets or stars, create a special kind of gravity effect called gravitomagnetism. This idea comes from comparing gravity to how electric charges work when they move. Just like moving electric charges create magnetic fields, rotating masses can create something similar in gravity.

The Earth has a very weak gravitomagnetic field because it spins slowly. Scientists tried to measure this with a special experiment called Gravity Probe B, but the effect is very small and hard to detect. Some very fast-spinning stars, called pulsars, might show stronger effects, but even then, the ideas we use to understand this get much more complicated.

Lack of invariance

Maxwell's equations look the same no matter where we view them from. But the GEM equations do not. This is because the amounts ρ_g and j_g do not act like a four-vector, unlike in electromagnetism. Instead, they are part of something called the stress–energy tensor, which is the source of gravity in general relativity.

Even if GEM might work in two different moving places linked by a special change, we cannot use one set of GEM values to find the other set, unlike with electromagnetism. Also, their ideas about what free fall looks like might not match. But GEM equations stay the same when we just move or turn our view, not when we look from more complex moving views. Maxwell's equations can stay the same under all these changes.

Gravitoelectromagnetism