Relativistic Physics: Principles and Applications
Elements of Relativistic Physics
In 1905, Einstein published his Theory of Relativity concerning motion in inertial frames. In 1916, Einstein expanded his theory to include non-inertial systems and gravitation, naming it the “General Theory of Relativity.”
Relativity in Classical Mechanics
We know that the trajectory of a body depends on the observer. For example, consider a plane that drops an object:
- The pilot observes that the trajectory of the falling object is straight (reference system O’).
- However, an observer at O sees the trajectory as a parabola.
Therefore, according to classical Newtonian mechanics:
- The trajectory and velocity of a body are relative, being dependent on the observer.
- Time is absolute since it is invariant for different reference systems O and O’.
Regarding the calculation of speed, Galileo stated that it is impossible to demonstrate by mechanical experiments if a reference system is at rest or moves with uniform rectilinear motion. In general, each observer can choose a coordinate system, which we call relative reference systems.
Transformations in Inertial Systems
An inertial reference system is at rest or moves with constant velocity. An event in the system is defined by O (x, y, z, t). To define a system O’, we have to use the equations of transformation:
a) If two observers are in the same reference system:
An event at P is determined by x = x’, y = y’, z = z’, t = t’.
b) If we have two reference systems OO’ with a constant distance OO’:
x’ = x – x0, y’ = y – y0, z’ = z – z0, t’ = t
Then we can measure distances in any reference frame O or O’.
Galilean Transformations
Suppose the observer O’ moves in the direction of the X-axis with constant velocity v, and at time t’ = t = 0, O and O’ coincide. Galileo’s equations are:
x’ = x – vt, y’ = y, z’ = z, t’ = t
Galileo’s equations are used for speeds (trains, airplanes, etc.) very small compared to that of light c, but not for high speeds similar to c.
Applications of the Galilean Transformation
a) The distance between two points is invariant in classical mechanics:
If O’ is moving with velocity v:
X1‘ = X1 – vt, X2‘ = x2 – vt, d’ = x2‘ – x1‘ = (x2 – vt) – (x1 – vt), d’ = x2 – x1 = d
Then the two observers measure the same distance.
b) The speed depends on the observer:
If two observers O and O’ measure the speed of an airplane:
For observer O: u = (x2 – x1) / (t2 – t1)
For observer O’: u’ = (x2‘ – x1‘) / (t2‘ – t1‘)
Where x1‘ = x1 – vt1, x2‘ = x2 – vt2, then u’ = ((x2 – vt2) – (x1 – vt1)) / (t2‘ – t1‘) = ((x2 – x1) – (t2 – t1)v) / (t2‘ – t1‘)
If the measurements are for t1 = t1‘, t2 = t2‘, it implies that u’ = u – v.
c) The acceleration is invariant under a Galilean transformation:
If the aircraft speed increases from A1 to A2 in O’, it implies u1‘ = u2 – v, u1‘ = u2 – v, u2‘ – u1‘ = (u2 – v) – (u1 – v) = u2 – u1. Then for O and O’, the two observers observe the same Δv at time t, so (u2‘ – u1‘) / t = (u2 – u1) / t, which implies that a’ = a.
Assuming that m is invariant mass, Newton’s second law is valid for all inertial reference systems.
Galileo’s Relativity Principle
The laws of physics (mechanics) are the same in all inertial reference systems.
The Problem of Electromagnetism
Einstein observed that the laws of electromagnetism would be valid in a reference system in which an observer travels at the speed of light because there would be a stationary electromagnetic field and no electric field. Then Einstein said that Maxwell’s equations of electromagnetism are in contradiction with the Galilean transformation.
A) Maxwell’s equations show that electromagnetic waves travel at a constant speed c, the speed of light in a vacuum.
In the reference system O: ux = c
For an observer at O’: ux‘ = ux – v = c – v
And this is in contradiction with the solution of Maxwell’s equations that O’ has ux‘ = c.
If the speed of light is different for O and O’, Maxwell’s equations are also different for each of these systems, and this is in contradiction with Galileo’s relativity principle. You could say that Maxwell’s equations were wrong, and Newtonian mechanics should be reviewed, as Newton’s laws do not vary in a Galilean transformation.
B) Maxwell’s theory predicts the existence of electromagnetic waves that propagate in a vacuum with speed c. Maxwell’s theory does not need a medium for wave propagation. This contrasts with mechanical waves (sound), which need a medium for propagation. To resolve this conflict, it was considered that light waves propagated in the ether, which exists even in a vacuum, so that they propagated similarly to mechanical waves in their midst. The ether was at rest and was the reference system for measuring the absolute speed of a mobile, and all objects moved through it. Then Maxwell’s equations were valid only in the rest frame of the ether. The speed of light in a vacuum c was the speed in the frame of the ether at rest. Drawing on the ether, the Galilean transformation applied to the speed of light is u’ = u – v = c – v.
All attempts to detect the presence of ether were unsuccessful. It was found that the speed of light was the same in all inertial systems.
Michelson-Morley Experiment
Michelson and Morley sought evidence of the existence of the ether. They measured the speed of the Earth relative to the ether using a half-silvered mirror. They always obtained negative results, indicating it was impossible to measure the absolute velocity of Earth relative to the ether. Today we know that the speed of light in a vacuum is equal to any reference system; that is, the speed of light in a vacuum is invariant.
Lorentz-Fitzgerald Contraction
They suggested that if it takes light the same time to travel the two arms of the interferometer, whether for or against the ether, they should have different lengths. The arm located in the direction of the Earth’s motion would experience a contraction in its length, given by:
l’ = l(1 – v2 / c2)½ (Fitzgerald-Lorentz Contraction)
v = velocity of the interferometer (Earth)
l = rest length of the ether
c = speed of light
Then all material bodies that move through the ether contract in the same direction of motion in a proportion of (1 – v2 / c2)½. The maximum speed of a body would be c.
Special Theory of Relativity
Einstein interpreted the failure of the Michelson-Morley experiment, indicating that the speed of light c is the same for all inertial systems. Therefore, Maxwell’s equations are true for all reference systems, and he rejected the Galilean transformation and the existence of the ether.
Einstein’s Postulates and the Lorentz Transformation
- The laws of physics are valid and have the same mathematical expression in all inertial reference systems.
- The velocity of light c is the same for all inertial systems, regardless of the source or observer’s speed.
By assuming that distance and time are not absolute, new equations were derived:
x’ = γ(x – vt), y’ = y, z’ = z, t’ = γ(t – (vx) / c2) (Lorentz Transformation)
where γ = 1 / (1 – v2 / c2)½
Consequences:
- If v = 0, then γ = 1, and the relativistic equations reduce to the nonrelativistic equations of Galileo.
- If v << c, then γ ≈ 1, and the Galilean transformations are valid. If v > 0.1c, relativistic equations must be used.
- The relativistic equations are called Lorentz transformations.
Relativistic Transformation of Velocity
Suppose the velocity of a point P in O is u; about O’, it will be:
u’ = (x2‘ – x1‘) / (t2‘ – t1‘), u = (x2 – x1) / (t2 – t1)
Substituting into the Lorentz equations:
u’ = (γ(x2 – vt2) – γ(x1 – vt1)) / (γ(t2 – vx2 / c2) – γ(t1 – vx1 / c2))
This leads to:
u’ = (u – v) / (1 – (uv) / c2) (Equation of velocity in the theory of relativity)
Consequences of the Lorentz Transformation
There are contradictions in length, time, etc., that differ between relativistic mechanics and classical mechanics.
Time Dilation
An interval of time measured in O will be greater than that measured in O’. A moving clock runs slower than a stationary clock. This is known as time dilation:
t = γt’ or t = t’ / (1 – v2 / c2)½
The term t’ is called proper time.
Length Contraction
The length of an object is defined as the length of that object in the reference system in which the object is at rest.
l’ < l or l = (1 / γ)l’
where l = l’ / (1 – v2 / c2)½
l’ = proper length (length measured in the mobile system)
Relativistic Mass
Einstein showed that the mass of a moving object increases:
m = γm0 or m = m0 / (1 – v2 / c2)½
m0 = mass of an object at rest
m = mass of an object moving at velocity v in O’
Equivalence Between Mass and Energy
Einstein published the consequences of the relativistic increase of mass. For small velocities v:
1 / (1 – v2 / c2)½ ≈ 1 + ½v2 / c2 + (3/8)v4 / c4 + (5/16)v6 / c6 + …
Then we can write that 1 / (1 – v2 / c2)½ ≈ 1 + ½v2 / c2
So, m = m0 + ½m0v2 / c2
The relativistic kinetic energy of a moving object is K = ½m0v2 = (m – m0)c2
The increase in mass is m – m0 = ½m0v2 / c2, which implies mc2 = m0c2 + ½m0v2
mc2 = total energy of a body
m0c2 = energy of the body at rest
If the kinetic energy is zero, then m0c2 = mc2, which is the total energy of a body at rest:
E = m0c2 (Einstein’s mass-energy equivalence)