Einstein's Special Theory Of Relativity
Einstein’s Special Theory Of Relativity; An Introduction
At the beginning of the last century it was believed that most of physics was well understood, there were just two seemingly small dark clouds, which obscured our otherwise impressive view of the physical world. These two problems related to the inability to detect the luminiferous ether and the statistical anomaly in black body radiation. These ‘small’ clouds on the horizon did however grow to gigantic proportion, casting a gloomy shadow over the physics community, which resulted in two great revolutions viz. Relativity and Quantum theory. [Figure #1] The conflict between Newton’s particle mechanics (as exemplified by the success of statistical thermodynamics) and Wave theory (as exemplified by Maxwell’s laws of electromagnetism) becomes evident when we study Black body radiation and its resolution resulted in the downfall of classical physics and birth of Quantum Theory! Another such conflict is ‘brought to light’ when we consider the propagation of electromagnetic radiation (light) and the Newtonian view of space and time, one that can only be resolved with Special Relativity. [Figure #2]. In his theory Einstein demolished the concept of the luminiferous Ether (the medium through which light waves were thought to propagate) and this article is a brief summary of the events and reasoning, which lead to this revolutionary theory.
According to Galilean relativity, the laws of physics are the same for all inertial frames. [These are defined by Newton’s first law of motion and correspond to observers that are moving at a constant velocity i.e not experiencing any acceleration]. With the advent of Special Relativity, the laws of physics were no longer valid under Galilean transformations but instead had to be Lorentz covariant and Maxwell’s equations of electromagnetism were unintentionally the first to be written in this form. The inability to detect the Ether was an indication that our understanding of space and time was flawed and eventually culminated in Einstein's general theory of relativity, which not only discarded the notion of an absolute space and universal time but also demonstrated that gravity was a manifestation of a curvature in the space-time continuum. Matter tells space and time how to warp and these in turn determine how test particles move (along geodesics of maximised space-time intervals, or proper time). Newton’s laws are Galilean covariant whereas Maxwell’s laws are Lorentz covariant. Maxwell’s equations lead directly to a wave equation of electromagnetism and hence the speed of light itself, is naturally a relativistic law of physics. Both Galilean and Lorentz transformations relate to inertial frames moving at constant velocity but the latter arises due to the ‘constancy of the speed of light’, being elevated to the status of an actual law of physics, something which was unknown to Galileo et al.
Einstein later developed his theory so as to be applicable even to non inertial frames (those that are undergoing acceleration). By making acceleration relative, Einstein completely undermined Newton’s belief in an absolute space, since in Newtonian mechanics the laws of physics are only valid in inertial frames and acceleration is not relative but absolute (which infers the notion of an absolute space). Here we will be concentrating on the special theory of relativity and will only briefly allude to the general theory, which will be dealt with more thoroughly in a future article.
FIGURE 1
Einstein’s Special Relativity (SR) resulted from his study of Maxwell’s Laws of Electromagnetism, while the need to extend this theory for non uniform motion (acceleration) and also to make it valid in gravitational fields, led to General Relativity {GR}. [We can therefore consider SR as being a limited case of GR]. Special Relativity explains the constancy of the speed of light in empty space far away from any matter, while GR explains the local constancy in a gravitational field. SR is mandatory when dealing with very high velocities comparable to the speed of light, while the incredible accuracy GR becomes evident when dealing with (large) gravitational fields. Let us now look again at the reasons why Einstein was driven to redefine our understanding of space and time.
As a boy Einstein imagined riding along a beam of light at the same speed ‘c’, whose waveform would therefore appear stationary. According to Maxwell’s theory there is no such thing, since he had shown that light is an oscillating electromagnetic field. The young Einstein imagined what he would see if he looked in a mirror which was travelling with him at the speed of light. Would he see his own reflection, or would he see nothing due to the fact that light from his face would never be able to catch up with the mirror and not therefore be reflected to his eyes. According to Newtonian physics, the light from his face would never reach the mirror if he travelled at the speed of light and he would no longer be able to see himself. Hence he would be able to say that he was travelling (in an absolute sense) at the speed of light, in contradiction to the principle of (Einsteinian & Galilean) relativity. So we need to recognise that the speed of light is the same (300 000 000 m/s) for all inertial frames, irrespective of their velocity. Einstein was the first to realise that the concept of an Ether was a myth, but that instead we needed to redefine the nature of space and time. This was Einstein’s earliest realisation that the speed of light must be constant for all observers and the need to elevate this fact to a physical law (which is more rigorously demonstrated by Maxwell’s equations), resulted in the relative nature of space, time and mass. Consider an observer moving at half the speed of light (relative to us). If he is approaching us and we shine a beam of light towards him, Newton & Galileo would say that it must pass him at one and a half times the speed of light, whereas if he were moving in the same direction as the beam, it would overtake him at just half the speed of light. However according to Maxwell’s laws of electromagnetism, the speed of light must be the same for all observers; so who is correct? Originally, most people would claim that Newton’s mechanics were correct but Einstein realised that it was Maxwell’s laws, which were valid (they were the first ever laws of physics to be Lorentz covariant - - - - space and time are relative)
Einstein was influenced by Immanuel Kant’s view, that space and time were products of our perception. In other words, what we know of the world conforms to certain a priori categories, which although we recognise by experience do not arise from experience. These categories (e.g. space and time) are laid down by the mind and turn sense data into objects of knowledge. “Space and time are modes by which we think rather than conditions in which we live” [EINSTEIN]
FIGURE 2
Newtonian physics has an absolute 3 dimensional space and a separate universal time. All observers ‘slice’ their world into the same sections of space and time. The pairs of events marked ‘a’ and ‘b’ in each space frame, represent events that are simultaneous to all observers, such as 2 people striking a match at the same time (although their spatial position and separation could change with time, as depicted in Figure 2). Here we have a Galilean Relativity, in which the known laws of physics are the same for all inertial frames (and all constant velocity motion is relative). However, although observers moving at different relative speeds share the same time, they have different distances. Galilean transformations do not however accommodate the laws of electromagnetism and this is an indication of the inadequacy of this erstwhile understanding of space and time. As we shall see, the laws of physics become of primary importance in Einstein’s theories and it these laws which indicate the nature of our personal space time, rather than the other way around. In some respect we may say that our personal space-time must transform in such a way, so as to tolerate these physical laws [This is particularly so in the final form of Einstein’s theory — General Relativity]
Galileo had replaced Aristotle’s view of a single absolute space through which objects moved over a period of time, with the notion of separate inertial frames connected by time. In this view all ‘smooth, uniform motion’ is relative and the laws of physics are the same for all such inertial frames. Einstein elaborated Galileo’s notion of relativity so as to include Maxwell’s laws of electromagnetism (SR) and later extended this symmetry principle (relativity) even further, so as to include non inertial frames (GR).
Note that in Figure #3, what actually occurs are not mere ‘rotations’ of the axes (which would be valid if all 3 axes were spatial), but rather more sophisticated Lorentz transformations, acting upon a pseudo-Euclidean manifold. {Both ‘x’ and ‘t’ coordinates are actually ‘rotated’ in towards each other, as depicted in the axes in the lower left hand corner.) The term ‘COVARIANT’ refers to the fact that the laws of physics looks the same all coordinate systems. Covariance means that both sides of an equation change in the same way, preserving the validity of the equation (whereas invariance means that nothing changes) The need for operational definitions and their primacy to perception, are very much in accord with the critical positivism of Mach, who had strongly influenced the young Einstein. His positivist epistemology regarded the relationships expressed in the curl and divergence of electromagnetic fields, as being an abstract property whose independent nature consequently made the ether redundant. By establishing operational definitions for all basic terms - in which concepts are described in terms of the behaviour of a physical system - he in effect overthrew the concept of absolute space and universal time in order to make our experience (laws of physics) invariant
FIGURE 3
By fusing space and time Einstein had achieved a unity comparable with that of Maxwell. The aesthetic appeal of a symmetrical field and his desire for greater unification was to remain with him for the rest of his life. It was Minkowski who was the first to fully recognise the need to geometrically unify space and time together [cf. Figure #5] and group theory allows us to study the symmetry properties of this manifold. As artists know, the apparent shape and dimensions of objects can change when viewed from different angles. However there are invariants, even though the actual projections along our length, width and height (x, y and z axes), are different in each case. For example, the distance between the diagonal corners of a 3 dimensional box is invariant, as indeed is its volume, so we would say that these properties are symmetrical under a rotational transformation. Now symmetry under translation leads to conservation of momentum and energy, while symmetry under rotation explains the conservation of angular momentum. Symmetry also allows unification of the fundamental interactions. Einstein was the first to realise that other transformations can also take place when an observer undergoes a boost in their velocity. He was therefore keen to establish what these changes are and what were the invariants (in 4D space-time but not space or time alone). Hence Einstein preferred the name ‘Invariance Theory’ rather than the ‘Relativity Theory’. Special Relativity expressed/restored symmetry under boosts in velocity (inertial frames), while his General Relativity (GR) went on to make the laws of physics covariant (symmetrical), under acceleration (non inertial frames) and produced a new understanding of gravity.
Maxwell’s laws predict a constant speed of light ‘c’, even for observers moving rapidly towards or away from light source (confirmed by Michelson & Morley and other experiments). This requires relative space and time! Maxwell’s laws were the very first to be Lorentz covariant and thus demonstrate the fallibility of the Galilean transformation. but the Lorentz transformation does reduce approximately to the Galilean transformation when the speeds involved are small compared to the speed of light. The Lorentz ‘gamma factor' is so close to unity for most terrestrial speeds, that SR effects are not noticeable in everyday experience The Lorentz transformation only become noticeable at speeds close to that of light, whereupon the gamma factor is significantly greater than unity and is a measure of how much length contracts and time dilates for a moving frame.
The significant differences between the transformations of SR compared to Galilean relativity, is that it is not restricted to simply rotations and translations and reflections but also boost transformations due to a change in relative velocities. All these together constitute what is referred to as the so called Poincare transformations, in which it is not just the 3 spatial axes that are transformed amongst themselves but also a space axis (in the direction of motion), is transformed (mixed) with the time axis. Hence different moving observers slice up their space and time differently as depicted in Figure #3 above. The result of this is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer (the blue axes in Figure #3), sets up a line of clocks that are all synchronised so they all read the same time (e.g. events ‘a’ and ‘b’), then another observer who is moving along the line at high speed (the red axes), will see the clocks all reading different times. Observers who are moving relative to each other observe different events as being simultaneous - - - - each observer will have their own plane of simultaneity.
Einstein regards space-time as a four-dimensional continuum in which observers travelling at different speeds slice their space and time into different ‘foliations’. The net effect of this four-dimensional universe is that observers who are in motion relative to you, seem to have time coordinates that lean over in the direction of motion, and consider things to be simultaneous that are not simultaneous for you. Also spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a rotation out of three-dimensional space So the events marked (in blue), ‘a’ and ‘b’ are simultaneous in the ‘moving’ (blue) frame but ‘a’ clearly occurs before ’b’ in the ‘Rest’ (red) frame. As an analogy, consider the statement “the chimney of the house across the road is directly behind a telegraph pole”. This may be true for me but not for a person standing a few feet away. Likewise in SR we have to accept the fact that people have significantly different ‘realities’ depending on their relative speed. So the bare statement that “these two events occurred at the same time”, is meaningless unless one adds “according to this observer”. [We will examine this loss of simultaneity in Figure #4 below.]
This loss of simultaneity is relevant to GPS systems since the notion of 2 signals sent simultaneously from 2 different orbiting satellites, is not simultaneous for the stationary observer receiving it on the ground and vice versa and this has to be corrected for, when estimating the global position of an observer. Hence observers will not agree upon the time interval between event a and b (which are actually regarded as simultaneous in the blue frame). Also note that SR states that there will be a time dilation effect to take into account because of the speed of the satellite, while GR states that there will be a counteracting speeding up of time due to the weaker gravitational field, which is of a greater magnitude than the SR effect.
Relativity tends to view space-time as single ’frozen ice block’ (containing both past present and future), in which individual observers are able to illuminate different slices according to the tenets of SR. This is in contrast to ‘the river of time’ and views which relate to the flow of time’s arrow. Einstein’s theories do however emphasise a perennial philosophical problem, in that his block view of space-time does not differentiate time into a past, present and future. In other words there is no ‘NOW’ to distinguish past from future or give a special significance to being in the present (this is known as the problem of transience). “For we convinced physicists, the distinction between past present and future is only an illusion however persistent”[EINSTEIN]
Now consider the Blue Inertial Frame. Suppose events ‘a’ and ‘b’ corresponds to the front and back of red’s ruler, coinciding SIMULTANEOUSLY, with the front and back of blue’s ruler (top left hand diagram). This defines the length of red’s ruler in blue’s frame. However according to the Red Frame, when the back of his ruler is in line with the back of blue’s ruler (event a), the front of blue’s ruler has not yet aligned with the front of his ruler (event b is in the future of event b). In other words red regards blue’s ruler, as being shorter than his (ac in top right hand diagram). Hence loss of simultaneity leads to different measurements of the length and time interval between events a and b but spacetime ‘interval ab ‘ is invariant.
Just as different people around the world have different perspectives on the time of day, orientation of the stars, different languages, alphabet and even numerals, we now need to appreciate that different observers may also have their own personal perspective on space and time; this is even more evident when we study General Relativity. As in life, where an individual is determined by the sum total of their experiences, in the physical universe, space-time is composed of 4 dimensional events and two events may be simultaneous to one observer but not to another moving inertial frame Henceforth space and time will be treated as a 4 dimensional (Minkowski) manifold, rather than an absolute 3 dimensional space with a distinct time, which flows universally. The human mind likes to slice up the 4D continuum into 3D of space and 1D of time (a computer for example, might be quite happy to think in terms of a 4D space-time). However Einstein was the first to realise that observers moving at different speeds, slice up their space in different orientations. The 2 axioms of SR are shown in Figure #4 where it is demonstrated that elevating the constancy of the speed of light to an actual law of physics, brings about a downfall of simultaneity and this in turn implies the relative nature of space and time. "Common sense are those layers of prejudice that are laid down before the age of eighteen” [EINSTEIN]
FIGURE 4
As observed by all 3 spaceships (who are at rest in deep space relative to each other), the light signals from the middle ship, arrives at the outer 2 ships at the same time (due to the second axiom above). This conflicts with the ‘moving’ observer, whose worldview is depicted above. Because he sees the 3 spacecrafts moving together with a velocity ’V’, he observes the light arriving at the rear spaceship first! Einstein realised that time must move at different rates for different observers, if light moves at the same rate for all observers.
An equivalent explanation is as follows. Consider a man at the centre of a moving railway carriage who strikes a match. From his point of view, the light from the match hits the front and back of the carriage simultaneously. However from the viewpoint of a man standing on the platform (who is aligned with the man when he strikes the match), the light from the match reaches the back of the carriage before the front. This is because the speed of light is the same for both men and during the time it takes the light to reach the ends of the carriage, the train has travelled a certain distance forward. The light therefore strikes the back of the carriage before the front. Note that even though the relative nature of space and time (encapsulated by the Lorentz transformations) is derived from the 2nd axiom (the constant speed of light demanded by Maxwell’s equation), the cause and effect is actually the other way around. That is, it is actually the (Lorentz) relativistic nature of space and time, that causes an upper limit to communications (namely the speed of light), as is demonstrated in Figure #6 below.
FIGURE 5
Figure #5 is a Minkowski space-time drawing of the previous gedanken (thought experiment) but with a spacecraft and flashing light bulb, instead of a train and match. It illustrates how the concept of simultaneity and hence time and length is relative, In any physical theory the endeavour is to make sense of observations. Different observers see events from different perspectives. If they differ in choice and direction of their coordinate axes, they give different coordinates to the same points, and so on. Yet the observers agree on certain consequences of their observations; in Newtonian physics and Euclidean geometry they agree on the distance between points that are measured simultaneously. Special relativity explains how observers in a state of uniform relative motion differ about lengths and times but agree on a quantity called the interval. In each case they are able to do so because the relevant theory presents them with a group of transformations that converts one observer's measurements into another's and leaves the appropriate basic quantities invariant.. With the help of Figure #5 we can debunk the ‘Ladder in the barn ‘paradox’
QUESTION: How can you fit a 5m ladder into a 4m barn and close both front and back barn doors?
ANSWER: If the 5m ladder is travelling at 3/5 of the speed of light, it would be contracted to 4m and it could then ‘fit into’ the 4m barn. [But what does the person at rest with respect to. the barn think?] In the second diagram above, if “MY SPACE” is that of the person at rest w.r.t the ladder, then events ‘a’ and ‘b’ correspond to the front and back of the ladder coinciding with those of the barn doors and it is clear that they do not occur at the same time. However according to the view of the person at rest w.r.t. the barn (represented by the oblique coordinates), they are simultaneous events and the ‘moving ladder’ is observed to fit inside the closed barn.. Hence there is no actual paradox, just a loss of simultaneity for the observer at rest w.r.t. the ladder but not for the observer at rest w.r.t. the barn..
Moving bodies shrink and moving clocks run slower (the length must reduce in proportion to time otherwise objects would not be able to agree upon their own relative velocity). As an object's speed increases, so does its Kinetic Energy which, according to Einstein’s famous equation E=mc², means that its mass also increases. [Historically it was the relativistic increase in mass, which allowed Einstein to derive his equation for mass-energy equivalence as a corollary of this relativity theory. Mass, length and time are intimately related in physical laws, hence all 3 will be affected by relativity theory and not just time and length alone.]
The object mentioned above which is acted upon by a force F, appears to us, to be ‘accelerating in slow motion’ as compared to the rest frame and the force has to be applied for longer, in order for the object to reach the velocity v, according to our observations. We therefore conclude that is inertial mass has increased by the same factor gamma that its clocks have slowed down. [In this thought experiment we must stipulate that the object is being accelerated in a direction at right angles to the motion of the moving frame]. Hence moving bodies also increase in mass, as accommodated for in particle accelerators, (this is necessary if the conservation of relativistic momentum is to be conserved).
Hence in summary;
1. Moving clocks run slower.
2. Moving objects contract in length.
3. Moving objects increase in mass
FIGURE 6
If several objects depart simultaneously from the same point, all at vastly different velocities, although they each travel through space at a different rate, they all travel at the same ‘speed’ through space-time and will all arrive on a circle in space-time simultaneously. In this manner we can say that we all move through space-time at the same rate – that of light! The combined speed of any objects motion through space and its motion through time is always precisely equal to the speed of light. [An object at rest in space has the maximum speed through time, while a beam of light has the maximum speed through space.] Figure #5 highlights the difference in the actual application of Lorentz as opposed to Galilean Relativity. If a woman travelling at close to the speed of light, decides to increase her speed by (say) 10 km per second, we do not observe such an increase. This is because due to time dilation, their second could be a month for us, while their km could be contracted to a centimetre for us. Hence relative to us, she increases her speed by just 10 cm per month! But what about light? Well as is shown in Figure #5, the Lorentz formula gives the correct answer and hence the speed of light is the same for all observers, (even if you are travelling in the opposite direction at a very high speed). You cannot catch up with a beam of light! Einstein was the first to see through the fiction of the ether. The next phase of the story involves Einstein’s struggle to reconcile Special Relativity with gravity and leads to an even greater paradigm shift in our understanding of space and time.