In the course of scientific research it is not uncommon for research into one subject to have a profound influence on another subject. In order to understand the developments which lead to a fundamentally new way of describing gravity, it is necessary to understand the developments in electromagnetism. Electromagnetism was the key to unlocking the true nature of gravity.
The words electric and electricity originate from the Greek word Elektron which means amber. The reason for this was that it was discovered that rubbing amber generates static Electricity which was originally thought to be a form of magnetism.
In 1785 the French physicist Charles-Augustin Coulomb published three papers where he first revealed the equation which is now referred to as Coulomb’s law. The equation described the force \(F\) between two charged objects with charges \(q_1\) and \(q_2\). \[F = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\] The constant \(\epsilon_0\) is the permettivity of free space, also known as the electric constant, with units \(C^2m^{-2}N^{-1}\). The coulomb constant \(k_e\) is defined exactly. \[k_e = \frac{1}{4\pi\epsilon_0} \equiv c^2 \times 10^7\] Where \(c\) is the speed of light.
Coulomb’s equation looks very similar to Newton’s equation of gravity. The coulomb constant is also very large. The unit of charge is named after Coulomb and is also very large. In practice most charges are a very small fraction of a coulomb.
The electric field \(\mathbf{E}\) is a vector which describes the force which would be exerted on a stationary charge \(q\). The units of the electric field are Newtons per Coulomb \(NC^{-1}\) or equivalently volts per metre \(Vm^{-1}\).
\[\mathbf{F} = q\mathbf{E}\]
The electric field at a distance \(r\) around a point charge \(q\) is: \[\mathbf{E} = \frac{q}{4\pi\epsilon_0 r^2}\mathbf{\hat{r}}\] The polarization field \(\mathbf{P}\) is the displacement of electric charges by an electric field within material. In a vacuum there are no charges to be displaced so the polarization field is zero.
The electric displacement field \(\mathbf{D}\) is defined in terms of the electric field and the polarization field. The units of the displacement field and the polarization field are coulombs per square metre \(Cm^{-1}\).
\[\mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P}\]
Gauss’ law is named after Carl Friedrich Gauss who formulated it. It relates the divergence of the electric displacement field to the electric charge density \(\rho\). It describes the electric field which comes out of a volume containing a density of charge.
\[\nabla\cdot\mathbf{D} = \rho\]
The magnetic field, or magnetic flux, \(\mathbf{B}\) is a vector quantity defining a magnetic field. It measured in Teslas named after Nikola Tesla.
The magnetization vector \(\mathbf{M}\) defines how strongly a region is magnetized. The magnetization of a permanent magnet is a constant related to the volume of the material. Its units are amperes per meter \(Am^{-1}\).
The magnetic field strength \(\mathbf{H}\) is related to the magnetic flux. The units are amperes per metre \(Am^{-1}\).
\[\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})\]
The constant \(\mu_0\) is the magnetic permeability of space.
The Lorentz force law is attributed to Hendrick Lorentz Although it had been derived earlier by Oliver Heaviside. It extends the force exerted by an electric field on a charged particle to include the effect of a magnetic field on the charged particle with charge \(q\) moving with velocity \(\mathbf{v}\).
\[\mathbf{F} = q\mathbf{E} + q\mathbf{v}\times\mathbf{B}\]
Where \(\times\) is the vector cross product which means that the magnetic component of the force is perpendicular to both the charge’s velocity and the magnetic field.
Michael Faraday formulated a law describing magnetic induction. It is stated as the induced electromotive force in any closed circuit is equal to the negative of the time rate of the magnetic flux enclosed by the circuit.
\[\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}\]
This is stating that if the magnetic field changes or the wire moves through a magnetic field then an electromotive force will be induced.
Ampere’s law states that an electric current \(\mathbf{J}\) creates a magnetic field.
\[\nabla\times\mathbf{H} = \mathbf{J}\]
James Clerk Maxwell was a Scottish physicist who revolutionized the scientific world. He is relatively little known outside of the scientific community, but he should be ranked alongside Newton and other great names. He is responsible for the second great unification of electricity and magnetism.
Maxwell built on the works of Gauss, Ampere and Faraday to produce what are now known as the Maxwell equations. His original work produced twenty equations. Oliver Heaviside refined these equations into four equations. These can be written in several different forms. The most familiar forms being the differential form.
\[\nabla\cdot\mathbf{D} = \rho\]
It states that the amount of electric flux emerging from the surface of any volume of space is equal to the total charge inside the volume.
\[\nabla\cdot\mathbf{B} = 0\]
This is similar to Gauss’ Law for electric fields. Note that the right hand side is zero. This means that there is no such thing as a magnetic charge. This effectively states that magnetic monopoles don’t exist.
PS: If magnetic monopoles should ever be discovered, this equation can easily be modified to accommodate them.
\[\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}\]
It states that a changing magnetic field induces an electric potential in a wire loop. The magnetic field can be changing in strength or a wire can be moving within a magnetic field. It also states that an electric current creates a magnetic field. This is an example of symmetry between electricity and magnetism.
This is the principle on which electric generators work.
\[\nabla\times\mathbf{H} = \frac{\partial\mathbf{D}}{\partial t} + \mathbf{J}\]
Maxwell added an extra term to the right-hand side to solve the problem that an electric current creates changing electric field. The equation now states that both an electric current and a changing electric field can create a magnetic field.
This is the principle on which electromagnets work.
In a vacuum where \(\mathbf{J} = 0\), Maxwell’s third and fourth equations have a symmetry which has an interesting consequence. A changing electric field gives rise to a changing magnetic field. Likewise, a changing magnetic field gives rise to a changing electric field. The two types of field are self-propagating. This effectively describes electromagnetic waves.
First of all, take the curl of the left-hand side of the third and fourth equations, using the mathematical identity defining the curl of a curl.
\[\nabla\times\nabla\times\mathbf{E} = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}\] \[\nabla\times\nabla\times\mathbf{H} = \nabla(\nabla\cdot\mathbf{H}) - \nabla^2\mathbf{H}\]
The first term on both right-hand sides is zero due to Maxwell’s first and second equations.
\[\nabla\times\nabla\times\mathbf{E} = -\nabla^2\mathbf{E}\] \[\nabla\times\nabla\times\mathbf{H} = -\nabla^2\mathbf{H}\]
If we take Maxwell’s third equation and apply the relationship between \(\mathbf{B}\) and \(\mathbf{H}\).
\[\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} = -\mu_0\frac{\partial\mathbf{H}}{\partial t}\]
Now take the curl of both sides.
\[\nabla\times\nabla\times\mathbf{E} = -\nabla^2\mathbf{E} = -\mu_0\frac{\partial}{\partial t}(\nabla\times\mathbf{H})\]
Substituting the fourth equation into the right hand side and reordering gives a wave equation. Using symmetry the same can be done for the magnetic field.
\[\nabla^2\mathbf{E} = \epsilon_0\mu_0\frac{\partial^2}{\partial t^2} \mathbf{E}\] \[\nabla^2\mathbf{H} = \epsilon_0\mu_0\frac{\partial^2}{\partial t^2} \mathbf{H}\]
These equations describe electromagnetic waves travelling at speed:
\[c = \frac{1}{\sqrt{\epsilon_0\mu_0}}\]
Where \(c\) is the speed of light.
\[c^2\nabla^2\mathbf{E} = \frac{\partial^2\mathbf{E}}{\partial t^2}\] \[c^2\nabla^2\mathbf{H} = \frac{\partial^2\mathbf{H}}{\partial t^2}\]
Maxwell realized that these equations describe all forms of electromagnetic radiation as alternating electric and magnetic fields travelling in free space at the speed of light.
Maxwell and others of his time believed that light was propagated through a medium called the luminiferous aether. When the Michelson-Morley experiment effectively proved that the luminiferous aether didn’t exist, a contradiction emerged between Maxwell’s equations and classical relativity.
One issue was that light in a moving frame of reference should have a different velocity due the the difference in velocity between the stationary and moving frame. Applying the motion to Maxwell’s equations gives the wavelength staying the same but the frequency changes.
Another contradiction emerges from a thought experiment. Take a long wire with an electric current passing through it. The current will generate a magnetic field according to Maxwell’s equations. Now consider a stationary electron near to the wire. Being stationary it will not experience a force from the magnetic field. Now consider what happens if the electron and wire are moving at a constant velocity. Now, the electron is moving through a magnetic field and will experience a force. As the laws of physics have to be the same for all observers, something is obviously wrong.