Introduction to electromagnetism

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Electromagnetism is one of the fundamental forces of nature. Early on, electricity and magnetism were studied separately and regarded as separate phenomena. Hans Christian Ørsted discovered that the two were related – electric currents give rise to magnetism. Michael Faraday discovered the converse, that magnetism could induce electric currents, and James Clerk Maxwell put the whole thing together in a unified theory of electromagnetism. Maxwell's equations further indicated that electromagnetic waves existed, and the experiments of Heinrich Hertz confirmed this, making radio possible. Maxwell also postulated, correctly, that light was a form of electromagnetic wave, thus making all of optics a branch of electromagnetism. Radio waves differ from light only in that the wavelength of the former is much longer than the latter. Albert Einstein showed that the magnetic field arises through the relativistic motion of the electric field and thus magnetism is merely a side effect of electricity. The modern theoretical treatment of electromagnetism is as a quantum field in quantum electrodynamics.

In many situations of interest to electrical engineering, it is not necessary to apply quantum theory to get correct results. Classical physics is still an accurate approximation in most situations involving macroscopic objects. With few exceptions, quantum theory is only necessary at the atomic scale and a simpler classical treatment can be applied. Further simplifications of treatment are possible in limited situations. Electrostatics deals only with stationary electric charges so magnetic fields do not arise and are not considered. Permanent magnets can be described without reference to electricity or electromagnetism. Circuit theory deals with electrical networks where the fields are largely confined around current carrying conductors. In such circuits, even Maxwell's equations can be dispensed with and simpler formulations used. On the other hand, a quantum treatment of electromagnetism is important in chemistry. Chemical reactions and chemical bonding are the result of quantum mechanical interactions of electrons around atoms. Quantum considerations are also necessary to explain the behaviour of many electronic devices, for instance the tunnel diode.

Electric charge

Coulomb's law tells us that like charges repel and opposite charges attract.

Electromagnetism is one of the fundamental forces of nature alongside gravity, the strong force and the weak force. Whereas gravity acts on all things that have mass, electromagnetism acts on all things that have electric charge. Furthermore, as there is the conservation of mass according to which mass cannot be created or destroyed, there is also the conservation of charge which means that the charge in a closed system (where no charges are leaving or entering) must remain constant. The fundamental law that describes the gravitational force on a massive object in classical physics is Newton's law of gravity. Analogously, Coulomb's law is the fundamental law that describes the force that charged objects exert on one another. It is given by the formula

${\displaystyle F=k_{\text{e}}\frac{q_1{q}_2}{r^2}}$

where F is the force, k_e is the Coulomb constant, q₁ and q₂ are the magnitudes of the two charges, and r² is the square of the distance between them. It describes the fact that like charges repel one another whereas opposite charges attract one another and that the stronger the charges of the particles, the stronger the force they exert on one another. The law is also an inverse square law which means that as the distance between two particles is doubled, the force on them is reduced by a factor of four.

Electric and magnetic fields

Electric field lines point from positive charges to negative charges.

 

The force exerted on a positive charge by an electric field (left) and a magnetic field (right) combine to give the Lorentz force.

In physics, fields are entities that interact with matter and can be described mathematically by assigning a value to each point in space and time. Vector fields are fields which are assigned both a numerical value and a direction at each point in space and time. Electric charges produce a vector field called the electric field. The numerical value of the electric field, also called the electric field strength, determines the strength of the electric force that a charged particle will feel in the field and the direction of the field determines which direction the force will be in. By convention, the direction of the electric field is the same as the direction of the force on positive charges and opposite to the direction of the force on negative charges. Because positive charges are repelled by other positive charges and are attracted to negative charges, this means the electric fields point away from positive charges and towards negative charges. These properties of the electric field are encapsulated in the equation for the electric force on a charge written in terms of the electric field:

${\displaystyle F=qE}$

where F is the force on a charge q in an electric field E.

As well as producing an electric field, charged particles will produce a magnetic field when they are in a state of motion that will be felt by other charges that are in motion (as well as permanent magnets). The direction of the force on a moving charge from a magnetic field is perpendicular to both the direction of motion and the direction of the magnetic field lines and can be found using the right-hand rule. The strength of the force is given by the equation

${\displaystyle F=qvB\sin \theta }$

where F is the force on a charge q with speed v in a magnetic field B which is pointing in a direction of angle θ from the direction of motion of the charge.

If there is no charge enclosed by a closed surface, then the amount of electric field flowing into it must exactly cancel with the electric field flowing out of it.

 

Because the flow of magnetic field out of a closed surface must cancel with the flow into it, magnets must have both North and South poles which cannot be separated into monopoles.

The combination of the electric and magnetic forces on a charged particle is called the Lorentz force. Classical electromagnetism is fully described by the Lorentz force alongside a set of equations called Maxwell's equations. The first of these equations is known as Gauss's law. It describes the electric field produced by charged particles and by charge distributions. According to Gauss's law, the flux (or flow) of electric field through any closed surface is proportional to the amount of charge that is enclosed by that surface. This means that the greater the charge, the greater the electric field that is produced. It also has other important implications. For example, this law means that if there is no charge enclosed by the surface, then either there is no electric field at all or, if there is a charge near to but outside of the closed surface, the flow of electric field into the surface must exactly cancel with the flow out of the surface. The second of Maxwell's equations is known as Gauss's law for magnetism and, similarly to the first Gauss's law, it describes flux, but instead of electric flux, it describes magnetic flux. According to Gauss's law for magnetism, the flow of magnetic field through a closed surface is always zero. This means that if there is a magnetic field, the flow into the closed surface will always cancel out with the flow out of the closed surface. This law has also been called "no magnetic monopoles" because it means that any magnetic flux flowing out of a closed surface must flow back into it, meaning that positive and negative magnetic poles must come together as a magnetic dipole and can never be separated into magnetic monopoles. This is in contrast to electric charges which can exist as separate positive and negative charges.

The right-hand grip rule for a straight wire (top) and for a coiled wire (bottom). Electrical current passed through a wire coiled around an iron core can produce an electromagnet.

The third of Maxwell's equations is called the Ampère–Maxwell law. It states that a magnetic field can be generated by an electric current. The direction of the magnetic field is given by Ampère's right-hand grip rule. If the wire is straight, then the magnetic field is curled around it like the gripped fingers in the right-hand rule. If the wire is wrapped into coils, then the magnetic field inside the coils points in a straight line like the outstretched thumb in the right-hand grip rule. When electric currents are used to produce a magnet in this way, it is called an electromagnet. Electromagnets often use a wire curled up into solenoid around an iron core which strengthens the magnetic field produced because the iron core becomes magnetised. Maxwell's extension to the law states that a time-varying electric field can also generate a magnetic field. Similarly, Faraday's law of induction states that a magnetic field can produce an electric current. For example, a magnet pushed in and out of a coil of wires can produce an electric current in the coils which is proportional to the strength of the magnet as well as the number of coils and the speed at which the magnet is inserted and extracted from the coils. This principle is essential for transformers which are used to transform currents from high voltage to low voltage, and vice versa. They are needed to convert high voltage mains electricity into low voltage electricity which can be safely used in homes. Maxwell's formulation of the law is given in the Maxwell–Faraday equation—the fourth and final of Maxwell's equations—which states that a time-varying magnetic field produces an electric field.

The electromagnetic spectrum

Together, Maxwell's equations provide a single uniform theory of the electric and magnetic fields and Maxwell's work in creating this theory has been called "the second great unification in physics" after the first great unification of Newton's law of universal gravitation. The solution to Maxwell's equations in free space (where there are no charges or currents) produces wave equations corresponding to electromagnetic waves (with both electric and magnetic components) travelling at the speed of light. The observation that these wave solutions had a wave speed exactly equal to the speed of light led Maxwell to hypothesise that light is a form of electromagnetic radiation and to posit that other electromagnetic radiation could exist with different wavelengths. The existence of electromagnetic radiation was proved by Heinrich Hertz in a series of experiments ranging from 1886 to 1889 in which he discovered the existence of radio waves. The full electromagnetic spectrum (in order of increasing frequency) consists of radio waves, microwaves, infrared radiation, visible light, ultraviolet light, X-rays and gamma rays.

The lab frame

The electron's rest frame

A further unification of electromagnetism came with Einstein's special theory of relativity. According to special relativity, observers moving at different speeds relative to one another occupy different observational frames of reference. If one observer is in motion relative to another observer then they experience length contraction where unmoving objects appear closer together to the observer in motion than to the observer at rest. Therefore, if an electron is moving at the same speed as the current in a neutral wire, then they experience the flowing electrons in the wire as standing still relative to it and the positive charges as contracted together. In the lab frame, the electron is moving and so feels a magnetic force from the current in the wire but because the wire is neutral it feels no electric force. But in the electron's rest frame, the positive charges seem closer together compared to the flowing electrons and so the wire seems positively charged. Therefore, in the electron's rest frame it feels no magnetic force (because it is not moving in its own frame) but it does feel an electric force due to the positively charged wire. This result from relativity proves that magnetic fields are just electric fields in a different reference frame (and vice versa) and so the two are different manifestations of the same underlying electromagnetic field.

Conductors, insulators and circuits

Conductors

The charges in a perfect conductor rearrange so that the electric field is always zero inside.

A conductor is a material that allows electrons to flow easily. The most effective conductors are usually metals because they can be described fairly accurately by the free electron model in which electrons delocalize from the atomic nuclei, leaving positive ions surrounded by a cloud of free electrons. Examples of good conductors include copper, aluminum, and silver. Wires in electronics are often made of copper.

The main properties of conductors are:

1. The electric field is zero inside a perfect conductor. Because charges are free to move in a conductor, when they are disturbed by an external electric field they rearrange themselves such that the field that their configuration produces exactly cancels the external electric field inside the conductor.
2. The electric potential is the same everywhere inside the conductor and is constant across the surface of the conductor. This follows from the first statement because the field is zero everywhere inside the conductor and therefore the potential is constant within the conductor too.
3. The electric field is perpendicular to the surface of a conductor. If this were not the case, the field would have a nonzero component on the surface of the conductor, which would cause the charges in the conductor to move around until that component of the field is zero.
4. The net electric flux through a surface is proportional to the charge enclosed by the surface. This is a restatement of Gauss' law.

In some materials, the electrons are bound to the atomic nuclei and so are not free to move around but the energy required to set them free is low. In these materials, called semiconductors, the conductivity is low at low temperatures but as the temperature is increased the electrons gain more thermal energy and the conductivity increases. Silicon is an example of a semiconductors that can be used to create solar panels which become more conductive the more energy they receive from photons from the sun.

Superconductors are materials that exhibit little to no resistance to the flow of electrons when cooled below a certain critical temperature. Superconductivity can only be explained by the quantum mechanical Pauli exclusion principle which states that no two fermions (an electron is a type of fermion) can occupy exactly the same quantum state. In superconductors, below a certain temperature the electrons form boson bound pairs which do not follow this principle and this means that all the electrons can fall to the same energy level and move together uniformly in a current.

Insulators

In a dielectric material, an electric field can polarise the material.

Insulators are material which are highly resistive to the flow of electrons and so are often used to cover conducting wires for safety. In insulators, electrons are tightly bound to atomic nuclei and the energy to free them is very high so they are not free to move and are resistive to induced movement by an external electric field. However, some insulators, called dielectrics, can be polarised under the influence of an external electric field so that the charges are minutely displaced forming dipoles that create a positive and negative side. Dielectrics are used in capacitors to allow them to store more electric potential energy in the electric field between the capacitor plates.

Capacitors

A parallel plate capacitor

A capacitor is an electronic component that stores electrical potential energy in an electric field between two oppositely charged conducting plates. If one of the conducting plates has a charge density of +Q/A and the other has a charge of -Q/A where A is the area of the plates, then there will be an electric field between them. The potential difference between two parallel plates V can be derived mathematically as

${\displaystyle V=\frac{Qd}{{\epsilon }_{0}A}}$

where d is the plate separation and ${\epsilon }_0$ is the permittivity of free space. The ability of the capacitor to store electrical potential energy is measured by the capacitance which is defined as $C=Q/V$ and for a parallel plate capacitor this is

${\displaystyle C=\frac{{\epsilon }_{0}A}{d}}$

If a dielectric is placed between the plates then the permittivity of free space is multiplied by the relative permittivity of the dielectric and the capacitance increases. The maximum energy that can be stored by a capacitor is proportional to the capacitance and the square of the potential difference between the plates

${\displaystyle E=\frac{1}{2}C{V}^2}$

Inductors

An inductor is an electronic component that stores energy in a magnetic field inside a coil of wire. A current-carrying coil of wire induces a magnetic field according to Ampère's circuital law. The greater the current I, the greater the energy stored in the magnetic field and the lower the inductance which is defined $L={\mathrm{\Phi }}_B/I$ where ${\mathrm{\Phi }}_B$ is the magnetic flux produced by the coil of wire. The inductance is a measure of the circuit's resistance to a change in current and so inductors with high inductances can also be used to oppose alternating current.

Other circuit components

Component	Main function	Schematic symbol
Resistor	Impedes the flow of current
Battery	Acts as a power source
DC voltage source	Acts as a source of direct current (DC), a constant current which points in one direction
AC voltage source	Acts as a source of alternating current (AC), a varying current which periodically reverses direction
Diode	Allows current to flow easily in one direction but not another
Capacitor	Stores energy in electric fields, stores charge, passes low frequency alternating current
Inductor	Stores energy in magnetic fields, resists change in current

Circuit laws

Kirchoff's junction rule (above):

I₁ + I₂ + I₃ = I₄ + I₅

Kirchoff's loop rule (below):

V₁ + V₂ + V₃ + V₄ = 0

Circuit theory deals with electrical networks where the fields are largely confined around current carrying conductors. In such circuits, simple circuit laws can be used instead of deriving all the behaviour of the circuits directly from electromagnetic laws. Ohm's law states the relationship between the current I and the voltage V of a circuit by introducing the quantity known as resistance R

Ohm's law: ${\displaystyle I=V/R}$

Power is defined as ${\displaystyle P=IV}$ so Ohm's law can be used to tell us the power of the circuit in terms of other quantities

${\displaystyle P=IV=V^2/R=I^{2}R}$

Kirchhoff's junction rule states that the current going into a junction (or node) must equal the current that leaves the node. This comes from charge conservation, as current is defined as the flow of charge over time. If a current splits as it exits a junction, the sum of the resultant split currents is equal to the incoming circuit.

Kirchhoff's loop rule states that the sum of the voltage in a closed loop around a circuit equals zero. This comes from the fact that the electric field is conservative which means that no matter the path taken, the potential at a point does not change when you get back there.

Rules can also tell us how to add up quantities such as the current and voltage in series and parallel circuits.

For series circuits, the current remains the same for each component and the voltages and resistances add up:

${\displaystyle V_{tot}=V_1+V_2+V_3+\dots R_{tot}=R_1+R_2+R_3+\dots I=I_1=I_2=I_3=\dots }$

For parallel circuits, the voltage remains the same for each component and the currents and resistances are related as shown:

${\displaystyle V_{tot}=V_1=V_2=V_3=\dots \frac{1}{R_{tot}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\dots I_{tot}=I_1+I_2+I_3+\dots }$

Speed of gravity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Speed_of_gravity 

 

Speed of gravity

Exact values
metres per second	299792458
Approximate values (to three significant digits)
kilometres per hour	1080000000
miles per second	186000
miles per hour	671000000
astronomical units per day	173
parsecs per year	0.307
Approximate light signal travel times
Distance	Time
one foot	1.0 ns
one metre	3.3 ns
from geostationary orbit to Earth	119 ms
the length of Earth's equator	134 ms
from Moon to Earth	1.3 s
from Sun to Earth (1 AU)	8.3 min
one light year	1.0 year
one parsec	3.26 years
from nearest star to Sun (1.3 pc)	4.2 years
from the nearest galaxy (the Canis Major Dwarf Galaxy) to Earth	25000 years
across the Milky Way	100000 years
from the Andromeda Galaxy to Earth	2.5 million years

In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In the relativistic sense, the "speed of gravity" refers to the speed of a gravitational wave, which, as predicted by general relativity and confirmed by observation of the GW170817 neutron star merger, is the same speed as the speed of light (c).

Introduction

The speed of gravitational waves in the general theory of relativity is equal to the speed of light in a vacuum, c. Within the theory of special relativity, the constant c is not only about light; instead it is the highest possible speed for any interaction in nature. Formally, c is a conversion factor for changing the unit of time to the unit of space. This makes it the only speed which does not depend either on the motion of an observer or a source of light and / or gravity. Thus, the speed of "light" is also the speed of gravitational waves, and further the speed of any massless particle. Such particles include the gluon (carrier of the strong force), the photons that make up light (hence carrier of electromagnetic force), and the hypothetical gravitons (which are the presumptive field particles associated with gravity; however, an understanding of the graviton, if it exists, requires an as-yet unavailable theory of quantum gravity).

How gravity works

A hypothetical graviton particle is emitted by any atomic particle. It travels through space until it collides with another mass or atomic particle. However, rather than being reflected backwards towards the emitter, the graviton particle 'slides' around the collided particle, and is re-emitted by the collided particle exactly 180 degrees relative to the incident angle. The collided particle is in effect a secondary emitter of the graviton particle. Because of Newton's laws, the collided particle will accelerate towards the original emitter particle. To a macroscopic observer, it appears as if the two atomic particles are exerting some attractive force between them. The mechanism whereby graviton particles slides, or orbits around a collided particle is similar to light reflection and refraction. The graviton's refractive index is such that rather than being reflected back (like a mirror) or refracted by some angle of incidence (like light through water), the graviton is transmitted exactly 180 degrees from the angle of incidence that the graviton hit the attracted mass particle.

This absorption and retransmission of graviton particles by a graviton-collided atomic particle follows the conservation of mass and energy laws, hence a mass can attract surrounding mass infinitely far away, in a spherical region around it.

As predicted by Einstein, these graviton particles travel at the speed of light. Hence, when energy is converted into mass, the gravitational effect of that newly created matter is detectable by dividing spatial radial distance by the speed of light.

Static fields

The speed of physical changes in a gravitational or electromagnetic field should not be confused with "changes" in the behavior of static fields that are due to pure observer-effects. These changes in direction of a static field are, because of relativistic considerations, the same for an observer when a distant charge is moving, as when an observer (instead) decides to move with respect to a distant charge. Thus, constant motion of an observer with regard to a static charge and its extended static field (either a gravitational or electric field) does not change the field. For static fields, such as the electrostatic field connected with electric charge, or the gravitational field connected to a massive object, the field extends to infinity, and does not propagate. Motion of an observer does not cause the direction of such a field to change, and by symmetrical considerations, changing the observer frame so that the charge appears to be moving at a constant rate, also does not cause the direction of its field to change, but requires that it continue to "point" in the direction of the charge, at all distances from the charge.

The consequence of this is that static fields (either electric or gravitational) always point directly to the actual position of the bodies that they are connected to, without any delay that is due to any "signal" traveling (or propagating) from the charge, over a distance to an observer. This remains true if the charged bodies and their observers are made to "move" (or not), by simply changing reference frames. This fact sometimes causes confusion about the "speed" of such static fields, which sometimes appear to change infinitely quickly when the changes in the field are mere artifacts of the motion of the observer, or of observation.

In such cases, nothing actually changes infinitely quickly, save the point of view of an observer of the field. For example, when an observer begins to move with respect to a static field that already extends over light years, it appears as though "immediately" the entire field, along with its source, has begun moving at the speed of the observer. This, of course, includes the extended parts of the field. However, this "change" in the apparent behavior of the field source, along with its distant field, does not represent any sort of propagation that is faster than light.

Newtonian gravitation

Isaac Newton's formulation of a gravitational force law requires that each particle with mass respond instantaneously to every other particle with mass irrespective of the distance between them. In modern terms, Newtonian gravitation is described by the Poisson equation, according to which, when the mass distribution of a system changes, its gravitational field instantaneously adjusts. Therefore, the theory assumes the speed of gravity to be infinite. This assumption was adequate to account for all phenomena with the observational accuracy of that time. It was not until the 19th century that an anomaly in astronomical observations which could not be reconciled with the Newtonian gravitational model of instantaneous action was noted: the French astronomer Urbain Le Verrier determined in 1859 that the elliptical orbit of Mercury precesses at a significantly different rate from that predicted by Newtonian theory.

Laplace

The first attempt to combine a finite gravitational speed with Newton's theory was made by Laplace in 1805. Based on Newton's force law he considered a model in which the gravitational field is defined as a radiation field or fluid. Changes in the motion of the attracting body are transmitted by some sort of waves. Therefore, the movements of the celestial bodies should be modified in the order v/c, where v is the relative speed between the bodies and c is the speed of gravity. The effect of a finite speed of gravity goes to zero as c goes to infinity, but not as 1/c² as it does in modern theories. This led Laplace to conclude that the speed of gravitational interactions is at least 7×10⁶ times the speed of light. This velocity was used by many in the 19th century to criticize any model based on a finite speed of gravity, like electrical or mechanical explanations of gravitation.

Figure 1. One possible consequence of combining Newtonian Mechanics with a finite speed of gravity. If we assume a Fatio/Le Sage mechanism for the origin of gravity, the Earth spirals outwards with violation of conservation of energy and of angular momentum. In 1776, Laplace considered a different mechanism whereby gravity is caused by "the impulse of a fluid directed towards the centre of the attracting body". In such a theory, a finite speed of gravity results in the Earth spiraling inwards towards the Sun.

From a modern point of view, Laplace's analysis is incorrect. Not knowing about Lorentz invariance of static fields, Laplace assumed that when an object like the Earth is moving around the Sun, the attraction of the Earth would not be toward the instantaneous position of the Sun, but toward where the Sun had been if its position was retarded using the relative velocity (this retardation actually does happen with the optical position of the Sun, and is called annual solar aberration). Putting the Sun immobile at the origin, when the Earth is moving in an orbit of radius R with velocity v presuming that the gravitational influence moves with velocity c, moves the Sun's true position ahead of its optical position, by an amount equal to vR/c, which is the travel time of gravity from the sun to the Earth times the relative velocity of the sun and the Earth. As seen in Fig. 1, the pull of gravity (if it behaved like a wave, such as light) would then always be displaced in the direction of the Earth's velocity, so that the Earth would always be pulled toward the optical position of the Sun, rather than its actual position. This would cause a pull ahead of the Earth, which would cause the orbit of the Earth to spiral outward. Such an outspiral would be suppressed by an amount v/c compared to the force which keeps the Earth in orbit; and since the Earth's orbit is observed to be stable, Laplace's c must be very large. As is now known, it may be considered to be infinite in the limit of straight-line motion, since as a static influence it is instantaneous at distance when seen by observers at constant transverse velocity. For orbits in which velocity (direction of speed) changes slowly, it is almost infinite.

The attraction toward an object moving with a steady velocity is towards its instantaneous position with no delay, for both gravity and electric charge. In a field equation consistent with special relativity (i.e., a Lorentz invariant equation), the attraction between static charges moving with constant relative velocity is always toward the instantaneous position of the charge (in this case, the "gravitational charge" of the Sun), not the time-retarded position of the Sun. When an object is moving in orbit at a steady speed but changing velocity v, the effect on the orbit is order v²/c², and the effect preserves energy and angular momentum, so that orbits do not decay.

Electrodynamical analogies

Early theories

At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Wilhelm Eduard Weber, Carl Friedrich Gauss, Bernhard Riemann and James Clerk Maxwell. Those theories are not invalidated by Laplace's critique, because although they are based on finite propagation speeds, they contain additional terms which maintain the stability of the planetary system. Those models were used to explain the perihelion advance of Mercury, but they could not provide exact values. One exception was Maurice Lévy in 1890, who succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light. However, those hypotheses were rejected.

However, a more important variation of those attempts was the theory of Paul Gerber, who derived in 1898 the identical formula, which was also derived later by Einstein for the perihelion advance. Based on that formula, Gerber calculated a propagation speed for gravity of 305000 km/s, i.e. practically the speed of light. But Gerber's derivation of the formula was faulty, i.e., his conclusions did not follow from his premises, and therefore many (including Einstein) did not consider it to be a meaningful theoretical effort. Additionally, the value it predicted for the deflection of light in the gravitational field of the sun was too high by the factor 3/2.

Lorentz

In 1900, Hendrik Lorentz tried to explain gravity on the basis of his ether theory and the Maxwell equations. After proposing (and rejecting) a Le Sage type model, he assumed like Ottaviano-Fabrizio Mossotti and Johann Karl Friedrich Zöllner that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. This leads to a conflict with the law of gravitation by Isaac Newton, in which it was shown by Pierre-Simon Laplace that a finite speed of gravity leads to some sort of aberration and therefore makes the orbits unstable. However, Lorentz showed that the theory is not concerned by Laplace's critique, because due to the structure of the Maxwell equations only effects in the order v²/c² arise. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low. He wrote:

The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light.

In 1908, Henri Poincaré examined the gravitational theory of Lorentz and classified it as compatible with the relativity principle, but (like Lorentz) he criticized the inaccurate indication of the perihelion advance of Mercury.

Lorentz covariant models

Henri Poincaré argued in 1904 that a propagation speed of gravity which is greater than c would contradict the concept of local time (based on synchronization by light signals) and the principle of relativity. He wrote:

What would happen if we could communicate by signals other than those of light, the velocity of propagation of which differed from that of light? If, after having regulated our watches by the optimal method, we wished to verify the result by means of these new signals, we should observe discrepancies due to the common translatory motion of the two stations. And are such signals inconceivable, if we take the view of Laplace, that universal gravitation is transmitted with a velocity a million times as great as that of light?

However, in 1905 Poincaré calculated that changes in the gravitational field can propagate with the speed of light if it is presupposed that such a theory is based on the Lorentz transformation. He wrote:

Laplace showed in effect that the propagation is either instantaneous or much faster than that of light. However, Laplace examined the hypothesis of finite propagation velocity ceteris non mutatis [all other things being unchanged]; here, on the contrary, this hypothesis is conjoined with many others, and it may be that between them a more or less perfect compensation takes place. The application of the Lorentz transformation has already provided us with numerous examples of this.

Similar models were also proposed by Hermann Minkowski (1907) and Arnold Sommerfeld (1910). However, those attempts were quickly superseded by Einstein's theory of general relativity. Whitehead's theory of gravitation (1922) explains gravitational red shift, light bending, perihelion shift and Shapiro delay.

General relativity

Background

General relativity predicts that gravitational radiation should exist and propagate as a wave at lightspeed: A slowly evolving and weak gravitational field will produce, according to general relativity, effects like those of Newtonian gravitation (it does not depend on the existence of gravitons, mentioned above, or any similar force-carrying particles).

Suddenly displacing one of two gravitoelectrically interacting particles would, after a delay corresponding to lightspeed, cause the other to feel the displaced particle's absence: accelerations due to the change in quadrupole moment of star systems, like the Hulse–Taylor binary, have removed much energy (almost 2% of the energy of our own Sun's output) as gravitational waves, which would theoretically travel at the speed of light.

Two gravitoelectrically interacting particle ensembles, e.g., two planets or stars moving at constant velocity with respect to each other, each feel a force toward the instantaneous position of the other body without a speed-of-light delay because Lorentz invariance demands that what a moving body in a static field sees and what a moving body that emits that field sees be symmetrical.

A moving body's seeing no aberration in a static field emanating from a "motionless body" therefore causes Lorentz invariance to require that in the previously moving body's reference frame the (now moving) emitting body's field lines must not at a distance be retarded or aberred. Moving charged bodies (including bodies that emit static gravitational fields) exhibit static field lines that bend not with distance and show no speed of light delay effects, as seen from bodies moving with regard to them.

In other words, since the gravitoelectric field is, by definition, static and continuous, it does not propagate. If such a source of a static field is accelerated (for example stopped) with regard to its formerly constant velocity frame, its distant field continues to be updated as though the charged body continued with constant velocity. This effect causes the distant fields of unaccelerated moving charges to appear to be "updated" instantly for their constant velocity motion, as seen from distant positions, in the frame where the source-object is moving at constant velocity. However, as discussed, this is an effect which can be removed at any time, by transitioning to a new reference frame in which the distant charged body is now at rest.

The static and continuous gravitoelectric component of a gravitational field is not a gravitomagnetic component (gravitational radiation); see Petrov classification. The gravitoelectric field is a static field and therefore cannot superluminally transmit quantized (discrete) information, i.e., it could not constitute a well-ordered series of impulses carrying a well-defined meaning (this is the same for gravity and electromagnetism).

Aberration of field direction in general relativity, for a weakly accelerated observer

Main article: Liénard–Wiechert potential

The finite speed of gravitational interaction in general relativity does not lead to the sorts of problems with the aberration of gravity that Newton was originally concerned with, because there is no such aberration in static field effects. Because the acceleration of the Earth with regard to the Sun is small (meaning, to a good approximation, the two bodies can be regarded as traveling in straight lines past each other with unchanging velocity), the orbital results calculated by general relativity are the same as those of Newtonian gravity with instantaneous action at a distance, because they are modelled by the behavior of a static field with constant-velocity relative motion, and no aberration for the forces involved. Although the calculations are considerably more complicated, one can show that a static field in general relativity does not suffer from aberration problems as seen by an unaccelerated observer (or a weakly accelerated observer, such as the Earth). Analogously, the "static term" in the electromagnetic Liénard–Wiechert potential theory of the fields from a moving charge does not suffer from either aberration or positional-retardation. Only the term corresponding to acceleration and electromagnetic emission in the Liénard–Wiechert potential shows a direction toward the time-retarded position of the emitter.

It is in fact not very easy to construct a self-consistent gravity theory in which gravitational interaction propagates at a speed other than the speed of light, which complicates discussion of this possibility.

Formulaic conventions

In general relativity the metric tensor symbolizes the gravitational potential, and Christoffel symbols of the spacetime manifold symbolize the gravitational force field. The tidal gravitational field is associated with the curvature of spacetime.

Measurements

For the reader who desires a deeper background, a comprehensive review of the definition of the speed of gravity and its measurement with high-precision astrometric and other techniques appears in the textbook Relativistic Celestial Mechanics in the Solar System.

PSR 1913+16 orbital decay

The speed of gravity (more correctly, the speed of gravitational waves) can be calculated from observations of the orbital decay rate of binary pulsars PSR 1913+16 (the Hulse–Taylor binary system noted above) and PSR B1534+12. The orbits of these binary pulsars are decaying due to loss of energy in the form of gravitational radiation. The rate of this energy loss ("gravitational damping") can be measured, and since it depends on the speed of gravity, comparing the measured values to theory shows that the speed of gravity is equal to the speed of light to within 1%. However, according to PPN formalism setting, measuring the speed of gravity by comparing theoretical results with experimental results will depend on the theory; use of a theory other than that of general relativity could in principle show a different speed, although the existence of gravitational damping at all implies that the speed cannot be infinite.

Jovian occultation of QSO J0842+1835 (contested)

In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had measured the speed of gravity indirectly, using their data from VLBI measurement of the retarded position of Jupiter on its orbit during Jupiter's transit across the line-of-sight of the bright radio source quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2 times the speed of light, which would be fully consistent with the theoretical prediction of general relativity that the speed of gravity is exactly the same as the speed of light.

Several physicists, including Clifford M. Will and Steve Carlip, have criticized these claims on the grounds that they have allegedly misinterpreted the results of their measurements. Notably, prior to the actual transit, Hideki Asada in a paper to the Astrophysical Journal Letters theorized that the proposed experiment was essentially a roundabout confirmation of the speed of light instead of the speed of gravity.

It is important to keep in mind that none of the debaters in this controversy are claiming that general relativity is "wrong". Rather, the debated issue is whether or not Kopeikin and Fomalont have really provided yet another verification of one of its fundamental predictions.

Kopeikin and Fomalont, however, continue to vigorously argue their case and the means of presenting their result at the press conference of the American Astronomical Society (AAS) that was offered after the results of the Jovian experiment had been peer-reviewed by the experts of the AAS scientific organizing committee. In a later publication by Kopeikin and Fomalont, which uses a bi-metric formalism that splits the space-time null cone in two — one for gravity and another one for light — the authors claimed that Asada's claim was theoretically unsound. The two null cones overlap in general relativity, which makes tracking the speed-of-gravity effects difficult and requires a special mathematical technique of gravitational retarded potentials, which was worked out by Kopeikin and co-authors but was never properly employed by Asada and/or the other critics.

Stuart Samuel also showed that the experiment did not actually measure the speed of gravity because the effects were too small to have been measured. A response by Kopeikin and Fomalont challenges this opinion.

GW170817 and the demise of two neutron stars

The detection of GW170817 in 2017, the finalé of a neutron star inspiral observed through both gravitational waves and gamma rays, at a distance of 130 million light years, currently provides by far the best limit on the difference between the speed of light and that of gravity. Photons were detected 1.7 seconds after peak gravitational wave emission; assuming a delay of zero to 10 seconds, the difference between the speeds of gravitational and electromagnetic waves, v_GW − v_EM, is constrained to between −3×10⁻¹⁵ and +7×10⁻¹⁶ times the speed of light.

This also excluded some alternatives to general relativity, including variants of scalar–tensor theory, instances of Horndeski's theory, and Hořava–Lifshitz gravity.