At first glance, the laws of electromagnetism appear to
violate Einstein’s Theory of Relativity, which states that it is
not possible to tell if you are moving or if you are standing still.
According to Maxwell’s Laws of Electromagnetism,
two particles with the same charge moving in the same direction
will exert an attractive magnetic force on each other.
This occurs only for moving charged particles,
and not for stationary charged particles.
From Sarah’s point of view, the two charged particles are moving,
and they will exert an attractive magnetic force on each other.
From Adam’s point of view, the two charged particles are standing still,
and there is no magnetic field and no magnetic force.
How is it possible to reconcile the two views?
Although Adam and Sarah will disagree about if there is a magnetic field present,
they will both agree that there are electric fields present.
All charged particles create electric fields, regardless of if they are moving.
Adam and Sarah will agree that the electric fields cause the two particles
to repel each other due to the fact that the two particles have the same charge.
In addition to the repulsive electric force,
Sarah will also see the attractive magnetic force.
Sarah will see the repulsive electric force
as stronger than the attractive magnetic force.
Therefore, she will still see the two particle move away from each other,
but she will see them move away from each other slower
than they would if there was no magnetic field present.
But, Sarah will not be able to use this as evidence to convince Adam that he is moving.
This is because Sarah will see Adam’s clock running slower than her own.
When Adam uses his clock
to determine how fast the particles are moving away from each other,
he concludes that it is at exactly the rate as if there was no magnetic field present.
Hence, all of Adam’s observations remain consistent with his belief that
there is no magnetic field present and his belief that he is standing still.
Einstein’s Theory of Relativity has therefore not been violated.
But let us try to violate Einstein’s Theory of Relativity in a
slightly more clever way by introducing gravity into the picture.
Suppose we prepare two charged spheres such that their electric repulsion
is exactly balanced by their gravitational attraction.
Adam will see that the forces are balanced and that the
distance between the two spheres is neither increasing nor decreasing.
As a result, Sarah must also see that
the distance between the two spheres is neither increasing nor decreasing.
But, how is it possible for Sarah to see all the forces balance
when she also believes that there is an additional magnetic force present,
which does not exist for Adam?
The answer is that there is a magnetic analogy to gravity.
The gravitational field that we normally think of around an object with mass
is analogous to the “electric” field around an object with charge.
We can also think of a new type gravitational field created by a “moving” mass
that is analogous to the “magnetic” field created by a moving charge.
Just as in the case of magnetism, this new type of gravitational field would exert
an apparent force perpendicular to the direction of the field
and perpendicular to the direction of motion.
Just as in the case of magnetism, this new type of gravitational field would exert
an apparent force perpendicular to the direction of the field
and perpendicular to the direction of motion.
As a result, in the case of our last example, Sarah will see all the apparent forces
exactly cancel each other out, and she will see that the two spheres
do not move toward or away from each other.
Adam and Sarah will disagree about whether or not
this new type of gravitational field is present, just as they will disagree
about whether or not there is a magnetic field present.
But this is OK, since we can’t actually observe these fields directly,
but only their effects on objects.
When observing the behavior of objects, there is no way for Adam and Sarah
to determine which of them is moving, and which of them is standing still.
If all the objects are moving slowly relative to the speed of light
and all the gravitational fields are weak, then the effects of gravity can be
approximated by equations very similar to Maxwell’s Equations for electromagnetism.
Just as the strength of the electric field around an object
depends on the amount of charge inside, the strength of the traditional
gravitational field around an object depend on the amount of mass inside.
But, the traditional gravitational field points “towards” the object, so that
whereas two objects with positive charge will repel each other,
two objects with positive mass will attract each other.
Just as magnetic fields are produced by moving charges,
the new type of gravitational fields are produced by moving masses
in the same way, but with the direction of the fields for positive charges
reversed from the direction for positive masses.
Just as the magnetic flux leaving a volume is always zero,
the new type of gravitational flux leaving a volume is also always zero.
When the two types of gravitational fields change with time, they induce each other
in exactly the same way as changing electric and magnetic fields induce each other.
When the two types of gravitational fields change with time, they induce each other
in exactly the same way as changing electric and magnetic fields induce each other.
This representation of gravity gives predictions that are only an approximation
to Einstein’s Field Equations, and this approximation remains valid only in cases
where there are no strong gravitational fields present
and all the objects are moving slowly relative to the speed of light.
On the other hand, Maxwell’s equations for electromagnetism remain valid
at any speed, but we need to take into account
the Relativistic effects, such as length contraction.
Length contraction affects electric and magnetic field lines just as it affects objects.
When Sarah thinks that field lines are closer together,
she will also think that there is a stronger field.
As a result, Sarah will think that both the electric and magnetic fields
perpendicular to the direction of motion are stronger
than they would be from classical physics alone.
Another relativistic effect is that as an object approaches the speed of light,
a force on it will produce a smaller acceleration than it would in classical physics.
Since this variable changes at a different rate depending on
if the force is perpendicular or parallel to the direction of motion,
the acceleration will also be different depending on if the force
is perpendicular or parallel to the direction of motion.
The end result of all these effects added together is that
it is never possible for Adam and Sarah to determine which of them is moving,
and which of them is standing still.