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Albert Einstein (c. 1905)
In the mathematics of special relativity, a light cone, is the path that a beam of light at a specified point in spacetime will follow through all directions of space. These are all of the events that can be reached by a light signal that is traveling to the future or coming to the past. There is thus, both a past and a future light cone. It also should be noted that the horizontal axis represents space and the vertical axis represents time. This is the temporal evolution of a beam of light in Minkowski spacetime!
Special relativity is a theory on the structure of space and time, or spacetime. Special relativity was introduced by Albert Einstein in 1905. The motivation for special relativity was an inconsistency between Newtonian mechanics and James Clerk Maxwell's theory of electromagnetism. Newton's theory was very successful at describing mechanical phenomenon, and Maxwell's theory had much success at describing the electromagnetic interaction. However, a problem had to do with the speed of light. Einstein would ask the question: what would a beam of light look like if you could catch up to it? Einstein's new theory of space and time was able to shed light on some of these questions.
Einstein's original paper was titled; “On the Electrodynamics of moving objects”. Special relativity breaks away from Newtonian mechanics, by replacing the Galilean transformations, with the Lorentz transformations, where space and time are interwoven into one continuum. Galilean transformations assume an absolute space and an absolute time. In the Lorentz transformations, each event that happens, happens at a distinct point in spacetime, and can be measured by an observer in each frame of reference. Special relativity is different from classical mechanics, in two main ways:
The speed of light is 186,000 miles per second.
The laws of physics are same for all inertial reference frames. Inertial frames are objects that move with respect to each other. This is the principle of relativity that was set out as early as the 17th century by Galileo. Physical laws are the same for all observers. Inertial frames of reference are environments free of acceleration. This is a reassertion of Newton's first law in a more general form. There is no such thing as an absolute velocity: only relative velocities.
The speed of light is a constant for all inertial reference frames. The speed of light, in a vacuum, is the same for every observer. This was a prediction by James Clerk Maxwell, that the speed of light is a constant. The speed of light will be the ultimate and maximum velocity in the universe. An experimental finding has been elevated to a fundamental principle of physics! The speed of light was a fundamental truth. There was no aether.
These were the most profound insights about nature since the time of Isaac Newton. This was a new picture of space and time that would diverge from Newtonian mechanics at velocities close to the speed of light.
Special relativity, is special, because, it is only applicable in situations where the effects of gravitation and the curvature of spacetime can be ignored. The kind of spacetime used in special relativity is known as Minkowski space and it is flat.
The idea of relativity, was not a new one, in fact, Galileo, in 1632, had proposed that the laws of motion are the same for all inertial frames of reference.
Another motivation for Einstein’s work on special relativity was the inconsistency between Newtonian mechanics and the equations of James Clerk Maxwell for electromagnetism. This was a theoretical implication of electromagnetism, there was a speed of light dependent on the medium that it propagates through. This is the second great unification for physics: electricity and magnetism by James Clerk Maxwell. James Clerk Maxwell, is going to bring together electricity and magnetism, as the same phenomenon. In so doing, the first classical theory of electromagnetism and of electromagnetic radiation is formulated. Maxwell, in 1865, published his demonstration that magnetic as well as electric fields, propagate as waves, through space, at the speed of light. Electricity, magnetism and light, are all manifestations of the same field.
Now, Albert Einstein, building on work of Henri Poincare and Hendrik Lorentz, was able to propose special relativity. Poincare is going to provide a mathematical framework for relativity, while Lorentz, is going to lay the fundamentals for Einstein's work.
Once it was established that light propagated as a wave, it was proposed that light would propagate through this "luminiferous aether". This would be similar to the way that sound travels through air. However, the aether had never been detected.
Einstein, is going to extend the work of Newtonian mechanics to include the speed of light. The speed of light was first observed in the Michelson-Morley experiment. The Michelson-Morley experiment was conducted in 1887 and compared light’s velocity in directions perpendicular to one another. The idea was that there was a “luminiferous aether”, a medium, that light would propagate through. However, the results of the Michelson-Morley experiment suggested that no such aether existed. In fact, the lack of evidence for the luminiferous aether was one of Einstein’s major motivations for relativity.
Albert A. Michelson and Edward Morley (1887) experiment
The lack of experimental evidence for the "aether" led to Hendrik Lorentz (see below) to develop a theory of electromagnetism in 1892. The theory was based on a completely motionless aether that could not be dragged by matter.
There were various pioneers who made significant contributions to applying the idea of 4-dimensional spacetime to the physical world. A little bit about the history of spacetime and special relativity:
Einstein and Lorentz
George FitzGerald
Hendrik Lorentz was a Dutch theoretical physicist, who laid much of the fundamentals for Einstein's theory of special relativity. In fact, the theory was originally called the Lorentz-Einstein theory.
His 1892 theory attempted to explain the Michelson-Morley experiment. It was at this time that he postulated the hypothesis of length contraction. Moving bodies contract in the direction of motion. The theory also assumed the existence of electrons that were strictly separated from the aether. This was an attempt to save the aether theory by Hendrik Lorentz and George FitzGerald who proposed that the earth and all measurements therein, such as in the Michelson Morley experiment, were actually compressed by the aether wind. The idea was that the Michelson-Morley experiment's results could be made consistent with the existence of an aether if the length of objects would contract in the direction of their motion. This was a proposed new property of the invisible aether by Lorentz and FitzGerald to reconcile the existence of the aether with the Michelson-Morley experiment: it could compress atoms as they passes through. This proposal would explain the negative result of the Michelson-Morley experiment. In this picture, the speed of light would have changed, however, this change could not be measured. The tool that you were using to measure would be compressed by the aether in just the right amount while the speed of light changed accordingly. This amount of shrinkage was calculated and led to what was known as the Lorentz-FitzGerald contraction. It was an electro-mechanical contraction of the atoms themselves. This will become central to special relativity. For Einstein, however length contraction will be for space itself, not the atoms that will be contracting. That being said, much of the physics community, Einstein included, were not enamored with the notion of the aether.
The Lorentz transformation, is named after Hendrik Lorentz and is the combined affect of length contraction and time dilation. These were the equations that would act to underpin Einstein's special theory of relativity. These were different from the Galilean transformations of classical mechanics. Since, now, time and space were no longer independent entities from one another. These Lorentz transformations were the transition of one inertial system to another, all the while, space and time are unified into one 4-dimensional continuum. It is the conversion of coordinates and time of an event to another coordinate and time of an event.
The key to Einstein's thinking is unification through symmetry. The way this worked in special relativity took a few steps:
Einstein had the picture that guided him through his research: running next to a beam of light.
This picture revealed a contradiction between Newtonian mechanics and Maxwell's equations.
Einstein was able to extract a constant principle: the speed of light.
Lastly, he put forth the Lorentz transformations, the symmetry that unifies space and time: rotations in space and time. Lorentz covariance was the phenomenon that the effects of physics retain the same form under the Lorentz transformations. Objects that maintain their form after a transformation are covariant.
Poincare also predicted the existence of gravitational waves in 1905.
The first to mathematically present Lorentz transformations in their modern symmetrical form was Henri Poincare. Poincare is regarded as one of the greatest French mathematicians of his time. He understood that the speed of light must be a constant in all frames. Poincare showed that Maxwell's equations retained the same form under a Lorentz transformation. Thus, providing the mathematical framework for special relativity.
Albert Einstein's 1905 paper was known as "On the Electrodynamics of Moving Bodies". However, the theory was not just for light, but for the universe. It was written on 31 hand written pages. It was an attempt at reconciling Maxwell's equations with the laws of mechanics. This was to account for some physical affects near the speed of light. This work would culminate in his special theory of relativity.
Herman Minkowski, the mathematics professor to the young (and perhaps chronically lazy) Einstein at the Polytechnic, once called Einstein a "lazy dog" for cutting his classes. Minkowski will later take an interest in his former student.
Hermann Minkowski, was Albert Einstein's former mathematics professor. He was the first to propose the geometrical formulation of special relativity. He proposed that special relativity could be understood best as four-dimensional spacetime. He will propose Minkowski spacetime. This will prove to be the flat and non-gravitational kind of spacetime that will be used in special relativity.
Minkowski took the mathematics of special relativity a step further. Minkowski tried to reformulate Einstein's idea that space turns into time and vice-versa the faster you move. When Minkowski put this idea into mathematical language, he found that space and time form a 4-dimensional unity. By re-writing Einstein's equations, Minkowski was able to link space and time into a 4-dimensional fabric.
Minkowski's work utilized the power of symmetry in an important way. Space and time were now different components of the same object. This was huge principle for the work of Einstein: unification through symmetry. Objects that maintain their form after a transformation are covariant. Herman Minkowski showed that the equations of Albert Einstein were covariant when formulated in 4-dimensional spacetime! All of the equations of relativity were allowed to be compressed into a remarkable and simple construction.
In the words of Einstein, had it not been for the work of Herman Minkowski, than relativity would have been "stuck in its diapers."
Herman Minkowski wrote:
"From now on, space and time separably have vanished into the merest shadows, and only a kind of union of the two will preserve any independent reality."
There are 7 consequences to special relativity that I want to discuss.
1) If two observers are in relative motion to one another, two events, that occur simultaneously, may not appear to be happening simultaneously. This is known as relativity of simultaneity. In other words, simultaneity is dependent on the reference frame of the observer. Events that seem to occur simultaneously for one observer, may not for another observer, and may appear to be separated by some significant periods of time.
An implication of relativity of simultaneity is that when two observers are in motion relative to each other, they will view time as running differently from each other. The effects of this are negligible at the scale of everyday experience, however, becomes apparent at relativistic velocities.
Whether two events appear to have occurred simultaneously, depends on the reference frame of the observer.
2) With the theory of special relativity the notion of a universal "now" is abandoned. Time is no longer universal, it is relative to the observer.
Observers now disagree on:
the time of events.
whether events occur simultaneously.
A clock that is moving, will be measured to tick slower, than the stationary clock of an observer. This phenomenon is known as time dilation. This is a difference in the measured elapsed time, by two different observers, due to a difference in their velocity, relative to each other.
A consequence of time dilation in special relativity: moving clocks, tick slower than stationary ones.
Moving clocks run slow in special relativity. However, they are not running slow in their own frame of reference. However, it will appear slow to an observer who is moving relative and not moving as fast.
Time dilation has been verified by many experiments and is now a well accepted concept in modern physics.
3) Objects that are moving, are measured to be shorter, in that direction that they are moving in. This is length contraction, that a moving object’s length, can be measured to be shorter than its proper length. The proper length, is of course, the length of the object at rest. Length contraction, or Lorentz contraction, is usually only noticed at relativistic velocities, these are significant fractions of the speed of light.
A visual example of length contraction: as an object's velocity, approaches the speed of light, it is measured to be shorter in length.
4) Maxwell's insight on the speed of light was that this measured velocity was how fast a beam of light traveled through the aether. This was fixed velocity that he measured for electromagnetic radiation. However, since there is no aether in special relativity, than what is the story behind the speed of light?
The speed of light is the upper limit on speed in the universe, according to special relativity. This principle has been experimentally established. The speed of light, is denoted as c and is a physical constant, that takes on the value of 186,000 miles per second. This is the maximum speed that all conventional matter and physical information can travel at. Such faster than light travel would lead to problems of causality, where, signals can be received before they are sent, the so-called: tachyonic antitelephone. There are situations where matter or physical information appears to move faster than the speed of light, however, they aren't. It’s impossible. The speed of light, is not only the speed of light, however, for all electromagnetic radiation traveling in a vacuum and for gravitational waves.
5) Another consequence of special relativity, given that, nothing can travel faster than light: gravity, cannot be an instantaneous force, as in the picture of Isaac Newton, however, gravitational waves, will propagate, at the speed of light.
Physical information and ordinary matter cannot move faster than the speed of light. The speed of light is about 186,000 miles per second. This is the velocity of all electromagnetic radiation and for gravitational waves. Nothing can go faster.
This picture demonstrates that light from the Sun, takes about 8 minutes and 17 seconds to reach the surface of the Earth.
6) It has been said that the most famous equation is: e=mc^2. This is the mass-energy equivalence formula of Albert Einstein. Mass and energy are equivalent under this formula. Mass and energy are alternative ways of describing the same thing! Anything with mass, has a corresponding energy. Anything with energy, has a corresponding mass. A particular energy, can be equivalent to its corresponding mass, as being calculated, as being multiplied by the speed of light squared. Energy and mass are transmutable. Also, this applies to all kinds of energy: potential, kinetic, electromagnetic, gravitational or nuclear.
This also explains why an object cannot be accelerated to the speed of light. When a body is accelerated it gains kinetic energy. Kinetic energy itself possesses mass according to this equivalence. To accelerate the body further becomes more difficult. Thus, the mass-energy would rise toward infinity as the speed of light is approached. That is why the speed of light cannot be reached.
According to the mass-energy equivalence formula, anything with mass, has a corresponding amount of energy and anything with energy, has a corresponding mass. This relationship can be calculated using the speed of light (c) squared.
7) Lastly, is the notion of mass, in special relativity or relativistic mass. When speaking of mass in special relativity, one is typically speaking of the rest mass of whatever it is one is looking at. The relativistic mass of an object increases as it approaches the speed of light. Your weight is not a constant. The faster you move, the heavier you get.
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