Special and General Relativity,

Transcription

1 Special and General Relativity, by Gary Oas, Education Program for Gifted Youth, Stanford University, 2009, revised 6/09. EPGY Special and General Relativity, by Gary Oas Lecture 1: Introduction, Why Relativity? Lecture 2: Relativistic effects via gedankenexperiments Lecture 3: Spacetime, spacetime diagrams, relativistic kinematics. Lecture 4: Geometry of Spacetime, metrics, relativistic effects. Lecture 5: Addition of Velocities, Spacetime Maps, Paradoxes, Causal structure. Lecture 6: Energy and Momentum in Special Relativity Lecture 7: Introduction to General Relativity Lecture 8: Curved Spaces, Effects of General Relativity Lecture 9: Schwarzschild Metric Lecture 10: Introduction to Black Holes. Lecture 11: Falling into a black hole, visualization near black holes. Lecture 12: Thermodynamics of black holes. Kruskal coordinates Lecture 13: Quantum Mechanics and Black Holes Lecture 14: Advanced topics Ventura Hall Stanford, CA oas@epgy.stanford.edu EPGY Summer Institute in Math & Physics 2001 [27] EPGY Summer Institute 2002 [14] EPGY Summer Institute 2002 (SRGR with calculus) [11] EPGY Summer Institute 2003 [21], 2004 [22], 2005 [22], 2006 [36], 2007 [29], 2008 [29], 2009 [24] NUS Physics Holiday Camp Dec 2004 [12] EPGY Holiday Camp Singapore Dec 2005 [14] 1

2 Lecture 1: Introduction, Why Relativity? EPGY Special and General Relativity, by Gary Oas What is relativity? The most straightforward way to explain what the theory of relativity is and how it relates to other areas of physics is in terms of geometry. The goal of this course is to impart the idea that relativity is nothing but geometry. When one first learns physics there is usually little discussion of geometry. The nature of space and time is taken to be self-evident; there are three independent dimensions of space (length, width, height) that is the same everywhere. Considering a basic physics problem one usually begins with defining three perpendicular axes to which reference is made. When finding the distance between points one employs the Pythagorean Theorem. This usually entails finding the length of the vector that begins at one point and ends on another. To find the length, one squares each component (v x 2, v y 2, v z 2 ), adds them, and takes the square root. As long as the directions (x,y, and z) form right angles at each point this is identically the Pythagorean Theorem (if this is not clear just consider two dimensions first). The other quantity taken to be obvious is the measurement of time. Time always flows into the future at the same rate for everyone. By taking these notions of space and time as evident, one has implicitly assumed the underlying geometry of space and time. Space is three-dimensional adhering to the rules of geometry set forth by Euclid (parallel lines never cross, etc.) and is hence called Euclidean geometry. Time is represented by a number line, a one-dimensional space where all points to one side of a point labeled NOW is called the past and those on the other side are called the future. In basic physics (mostly set forth by Isaac Newton and hence called Newtonian or classical physics) three-dimensional space is unrelated to time they are distinct entities. The modern view of relativity is one that modifies the previous statements. By insisting upon consistency within all areas of physics for all observers one is forced to use a different geometry underlying physics. First, one cannot consider space and time as separate entities but as belonging to a four-dimensional geometrical structure we call spacetime. The geometry of spacetime is found not to be Euclidean but something else. Getting comfortable with this new geometry is what makes relativity seemingly difficult to understand. This new geometry also introduces strange phenomena that seem counterintuitive and strange, but have all been experimentally verified many times over. Why it is difficult for us to comprehend is that these effects are only significant for motions close to the speed of light a realm that we do not have everyday experience with. Once the geometry of spacetime is accepted one proceeds in developing physics just as before. Since the underlying geometry influences how things are measured, it will effect all realms of physics. Thus one must treat physical notions with care. It is found that many, if not all, principles of Newtonian physics must be modified or abandoned. Again, the point of this course is that relativity is nothing but geometry. This course will approach the topic in the following manner: The first order of business (and probably the most difficult part to grasp) is to understand why things need to be changed in the first place. The first part of the course will gradually introduce the basic, bizarre, consequences of the new geometry. Beginning with simple thought experiments, then diagrammatic methods, and finally in terms of more formal geometry, the way that simple measurements are to be related will be understood. This portion of the course will end with a thorough exploration of the nature of spacetime and how mechanics changes. The second part of the course will focus on gravity and how it relates to the geometry of spacetime. What will be found is that gravity is the geometry of spacetime. Indeed, mass and energy will be seen to change the geometry of spacetime nearby. This is the theory of General Relativity. The last part of the course will explore areas where general relativity plays a key role: notably black holes. We will end the course discussing modern ideas and unresolved problems. The most general outline of this course would go as follows: Why did the theory arise in the first place? (1 st day). What are the basic notions of the theory of special relativity and what does it predict. How did the general theory come about? What are the basic notions of the general theory and what does it predict. The general theory introduces us to two new realms of physics black hole physics and cosmology. We will end the course discussing modern ideas and unresolved problems. 2

3 Where does the theory of relativity fit within the context of physics as a whole? Classical Physics: Mechanics (Newton s laws, Energy, Momentum, Universal Law of Gravitation) Wave Mechanics (simple harmonic motion, sound). Fluid Mechanics (Bernoulli s law, Archimede s Principle) Thermodynamics (4 laws of thermodynamics, energy, work, heat, entropy) Electricity and Magnetism (Coulomb s law, Electric fields/potential, Gauss Law, Ohm s law, Circuits, Magnetic forces/fields, Induction, Maxwell s equations, EM waves, light) Optics (Geometric optics, lens systems, waves, interference, diffraction) Relativity: Special relativity (kinematics: time dilation, length contraction, spacetime interval invariance, dynamics: momentum, energy, collision, E&M, fluids) General relativity (gravitation as curved space, Schwarzschild solution, black holes, cosmology). Quantum Mechanics: Atomic physics, spectra, energy levels, photoelectric effect, Bohr s model, wave functions, Heisenberg uncertainty principle, Schrödinger equation, Hydrogen atom, Atomic, molecular, and nuclear physics. Quantum Field Theory: Elementary particles, Feynman rules, S-matrix, the Standard Model, Higg s physics Quantum Gravity: Quantum Mechanics + General Relativity, no known theory yet, string theory, loop theory, M theory, Theory of Everything, quantum aspects of black holes. The theory of relativity often falls under the classification of modern physics even though it is almost 100 years old. However, this theory is the underpinning of many of the recent ballyhooed exotic theories; superstring theory, quantum gravity, loop quantum gravity, M-theory, etc. To begin to understand these extremely sophisticated ideas, a sound foundation in special and general relativity is a must. The other theory that is the basis of all of the new physics is quantum mechanics. Quantum mechanics ( years old) is a mathematically complex, extremely non-intuitive theory, the meaning and philosophy of which is still not understood. Without quantum mechanics, most of today s technology would not be possible. To understand the basic building blocks of nature, it becomes necessary to combine quantum mechanics and special relativity (into a theory called quantum field theory (QFT)). QFT is behind the development of the Standard Model, to date our best interpretation of the forces and particles of nature. However the Standard Model has one omission general relativity. Without its inclusion, it can not be a theory that can describe the entire universe. The earliest attempts to include general relativity with quantum mechanics met with failure. To this date, no other attempts have been successful. The search for a quantum theory of gravity is the primary theoretical goal of physics. The exotic theories discussed above are all attempts to carry this out. While none have been demonstrated to be correct, they have not been eliminated yet. The problem of developing a unified theory of all forces, including gravity, is of such difficulty that most agree it will require a revolution in thinking about our universe. Such a revolution may make the early revolutions of relativity and quantum theory seem trivial. 3

4 Why Relativity? To begin we will review classical mechanics. You should already be familiar with most of what we will discuss in the beginning of this lecture. Physics is the study of how objects behave and interact. In order to provide an unambiguous quantitative description of nature it is necessary to define the basic observations that are made, position and time. Other measurements are abstractions based on the position and time measurements. Assigning each object a position and a time, the goal is to devise a scheme to predict where the objects will be at a later time. These are the two basic measurements that are made. In order to relate positions and times of several objects a coordinate system is constructed. Without the coordinate system there is no way to construct meaningful statements relating different objects. Though a coordinate system is crucial to make a quantitative description, it in no way affects the motion or interaction of the objects. It is a construct removed from the real world. Any properly defined coordinate system is capable of describing the system. In mechanics, a properly defined coordinate system is one that has three perpendicular spatial directions, (usually called (x, y, and z)), and a zero for time. Locating the origin and orientation of the axis is sufficient to make quantitative measurements. Of course the units of measurement ought to be consistent. It is impractical to have one spatial direction measured in feet and another in kilometers. Position and time may be the fundamental observations made but they are not the most practical to use for relating measurements. For two different observers measuring the position of two objects the values may not come out the same. In order to relate measurements in different frames, it is best to reference positions in relation to other physical objects (not necessarily to your non-physical coordinate system). Observe that in Alice s and Bob s frame the positions of the two objects are not the same in the two frames. However, the magnitude of the displacement, or the distance, between the objects is the same for each (even when one frame is in motion with respect to the other). We will come back to this later in the next lecture. To catalogue positions of objects it is necessary to derive mathematical relations describing their movement, which can only be done once a coordinate system is established. This goes by the name kinematics. To derive these relations the concepts of displacement, speed, velocity, and acceleration need to be defined in terms of positions and times. It is assumed you are thoroughly familiar with these quantities. For motion that is at most moving with constant acceleration, the following equations can describe the trajectories of an object in one frame of reference. r = r 0 + v 0 t + 1 at 2 2 v = v 0 + at v = x t a = v t where r 0 and v 0 are the position and velocity at some fixed time (say t = 0).These equations describe the motion of an object which is not interacting with any other objects. To describe this motion a coordinate system is set up with a zero for time defined. This is usually done from your own perspective. You might choose the origin at your feet and look at your watch and designate t = 0 s at some instant. But if there is another making observations then that person might choose another coordinate system and it is important to relate the two observations. 4

5 Scenario 1: Two reference frames at rest with respect to each other. EPGY Special and General Relativity, by Gary Oas If Alice and Bob are at rest with respect to each other but are separated in space by a vector R, and they choose different times to set t = 0 then it should be clear that they each measure the same speed and magnitude of acceleration for a nearby object. However the velocities and accelerations will be different. This is because t = t and r = r. To relate Bob s measurement to Alice s we have: r ' = r R or r '+ R = r 0 + v 0 t v ' = r ' t = r t R t = r t = v at 2 å Scenario 2: Two frames moving at constant velocity with respect to each other. If Bob s reference frame is moving with respect to Alice s then things are slightly more complex. Consider Bob moving with constant velocity V with respect to Alice. Now we have the following, r ' = r R(t) or r '+ R(t) = r 0 + v 0 t + 1 at 2 2 v ' = r ' t = r t R(t) = v V t a '= v ' = v t t V t = v t = a where V is a constant vector in time representing the relative motion of Alice and Bob. 5

6 6 EPGY Special and General Relativity, by Gary Oas Scenario 3: Two reference frames accelerating with respect to each other. Lastly, if Bob is undergoing constant acceleration with respect to Alice then not even the accelerations of the two objects will agree for the two observers. Newton, and Galileo before him, thought that accelerated motion was unnatural, that the true nature of trajectories was either to be at rest or moving with constant velocity. For an acceleration to occur some outside entity, or force, had to be operating on the object. (Note that Aristotle thought that only objects at rest were the natural trajectories of objects, he thought a force was required to move an object at a constant velocity.) Newton s Laws To include interactions between objects we introduce Newton s laws. Newton s three laws describe how motions are related to forces and inertia. 1) Every body continues in its state of rest or of uniform speed in a straight line unless acted on by a nonzero net force. 2) The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object. ( F = ma, or more precisely F = (m v) ). t 3) Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first. Inertial Reference Frames We have seen that in considering two reference frames that are at rest with respect to each other, or moving at a constant velocity with respect to each other, the observed accelerations of an object will be the same. When two frames accelerate with respect to each other then the observed accelerations will be different. Thus, if in one frame one measures a force F acting upon an object with mass m and observes an acceleration of a = F/m, then Newton s 2 nd (and 1 st ) are valid. But note that in the frame which is accelerating with respect to the first the acceleration will not satisfy Newton s laws, a F/m. Forces will be the same in the two frames. Consider a loaded spring, in both frames F = -kx and k and x will be the same in both frames. Frames for which Newton s equations are valid are called Inertial Reference Frames, or IRFs for short. Frames for which Newton s laws do not hold are called non-inertial Reference Frames. Now, we stated that for the frame which is accelerating with respect to the first Newton s laws do not hold, thus we might conclude that any accelerating frame is a non-irf. Generally this is true, but when we come to general relativity we will see that we need to be very careful about this relation. Keep it formal, an IRF is a frame where Newton s laws are valid. With Newton s laws and the kinematical equations you can describe motions of objects and the interactions among them. You have, no doubt, done many problems along these lines and this will not be reviewed here. In mechanics there are generally two different types of forces discussed, gravity and contact forces. The gravitational force is often introduced on the surface of the Earth as a force which always produces the same acceleration for objects which are dropped. However you know that this is only an approximation and that Newton s Universal Law of Gravitation (NUG) gives the correct form, F = G M M 1 2 ˆr 2 12 r 12 As was discussed in the brief introduction to electricity and magnetism, the contact forces discussed in elementary mechanics courses are actually due to electric forces. Electricity and magnetism are described by Maxwell s equations developed in the 1860s. The other two fundamental forces of nature the strong and weak force-- only occur over subatomic ranges. We will not be interested in these forces and will not discuss them further. Newton s Universal Law of Gravitation is an extremely successful theory. Most spacecraft trajectories are calculated using the NUG to this day. However, towards the end of the nineteenth century some small discrepancies between observations and NUG became apparent. These we will leave off until we discuss General Relativity.

7 Electromagnetic Theory and Newtonian Physics On another front there were some discrepancies between classical mechanics and Maxwell s theory. In the last decades of the nineteenth century, Maxwell s theory had been proved to be a highly accurate and successful description of electromagnetic phenomena. However, the physical basis of electromagnetic waves was not well understood. Up to that point it had been known that all wave phenomena propagate through some medium: wave on a rope, water waves, etc. In Maxwell s theory it was not entirely clear what media the wave is propagating through. At the time, the accepted idea was that there was some entity through which EM waves propagate. The ether was introduced as the substance through which EM waves propagated. The ether was thought to permeate the entire universe. No one had ever observed this ether and its physical properties were debated at length. Several experiments were performed to attempt to measure or observe the ether. None were more famous than those performed by A. A. Michelson and E.W. Morley, now known as the Michelson-Morley experiment (M-M). This experiment attempted to measure the change in the speed of light as the Earth passed through the ether at different relative velocities. Recall that Maxwell s equations tell us that EM waves (light) travel at c = 3x10 8 m/s. At the time this was taken as the speed of light propagating through the ether. If the Earth, (along with the experiment), traveled at some speed relative to the ether, then the observed speed of light in the laboratory would differ from c. The M-M experiment is an optics experiment and here is a very brief description of the experiment (you will see more of this experiment in your problem set). The M-M experiment (a modern version of it) is based on a device called an interferometer. The modern version has a laser shining a coherent beam of light of constant wavelength into a beam splitter. A beam splitter splits the light into perpendicular paths. The light travels down each arm, reflects off a mirror, and returns it to a beam splitter, where it is rejoined with the other beam. See figure. The recombined beam is observed in the fourth arm. [For the following please see the mathematical appendix for a very brief introduction]. Since the light wave has only one wavelength and starts at the same phase at the beam splitter (same point in the wave), we see that the two waves that return may be out of phase depending on the length of L1 & L2. It is possible to adjust L1 & L2 to be very close using the interferometer and we will assume for simplicity that L1 = L2. If this is the case, (if the speed of the wave is the same in both arms), then the two beams will arrive back at the beam splitter in phase (peaks will match peaks). When the two waves are added at the beam splitter the combined light will undergo constructive interference and a bright point will be observed at the center of the pattern. Now consider if one of the mirrors is moved by 1/4 of a wavelength (on the order of a few hundred nanometers). So that the total difference between the two paths is 1/2 of a wavelength. Then the two waves will undergo destructive interference and a dark spot will be observed in the center. Returning to the experiment. Let s say the Earth is passing through the ether in the direction of arm 2. Then along this direction the wave is traveling at c = c - v upstream and c = c + v downstream. The beam traveling along arm 1 will experience a sideways deflection. 7

8 (Picture a boat traveling across a river with a strong current. It will have to aim slightly upstream to travel straight across). We see that the two beams will arrive at different times and a phase difference will be observed. To show the phase difference the entire experiment is rotated so that the ether wind travels down the other arm. The phase should shift as this is done and the center observed spot should change from light to dark a number of times while being rotated. The result? No phase shift was ever observed. The experiment was performed several times throughout the year to measure several different relative velocities. The conclusion of the experiment was that there was no ether. Various explanations were attempted, (e.g. the Earth dragged the ether along with it so v=0 on the surface), but all were found to be lacking. A few things were known about light that demonstrated that it was different than ordinary objects. It had been known that light emanating from an object travels at the speed of light. To contrast, consider Bob on a railroad car traveling with a constant speed V and he throws a ball towards the front of the train at speed v (as he sees it). To Alice, the ball will appear to travel at V+v. Now hand Bob a laser and have him shine it forward. Now Alice will observe it to travel at c (and Bob at c-v). This was concluded by astronomical observations of binary stars (James Bradley in 1728). If the speed of the light emanating from the stars was added to or subtracted from the star s orbital speed, then a non-uniform orbit would be observed (see animation). Considering again the example with Bob and the laser you might think that there is something strange about this result, and indeed there is. To explain why there is something fishy here, let s return to the Michelson-Morley experiment. The M-M experiment demonstrates that there is no ether through which light propagates. But there is something else. Even though there is no ether the arms are still traveling at different speeds relative to the universe. Without the ether, light waves were thought to propagate with respect to the universe as a whole, the one state of rest with which all motion is in respect to. There were several philosophical problems with this. Maxwell s equations state that light propagates with c = 299,792,458 m/s (or just 3x10 8 m/s), but with respect to what? In which reference frame does this occur? This one reference frame then is surely a very special reference frame. All other reference frames are not as special and you can always determine how your reference frame is moving to this special one by measuring the speed of light in your frame and then comparing with 3x10 8 m/s. Also since c = 3x10 8 m/s only occurs in this one special frame, then Maxwell s Equations are not valid in the others. Nowhere in Maxwell s theory is there a discussion of relative velocity. There seems to be a problem here. This problem simply stems from this special speed which is at the core of Maxwell s Theory. Recall that c is defined as µ ( ) 1 2 and 0 are merely constants that do not depend upon speed. There is no reference to any special speeds in mechanics. Some tried to explain this discrepancy and one very interesting attempt was first put forward by Fitzgerald in 1889 and then in detail by L Lorentz in In order to explain this effect he proposed that the arm 1 v2 along the direction of travel respect to this universal frame shrank by a factor of, where v was the speed 2 c of the experiment with relative to this special frame. This gave a phenomenological description of the effect but and µ 0

9 there was no physical basis proposed (it also still relied on the existence of the ether). It will be seen later that this is the correct description of length contraction obtained by Einstein. Enter Albert Einstein When Einstein was young he constantly tried to figure out an answer that just about everyone asks at one time or another. What would you observe if you traveled at the speed of light? Let s try to see by proposing a gedankenexperiment. A superfast spaceship is developed that can travel at the speed of light. Bob gets in and travels alongside a laser beam fired from Earth. To see what happens let s review how electromagnetic waves propagate. In deep space there are no charges to create electric (E) or magnetic (B) fields. The last two of Maxwell's equations tells us that a time varying magnetic field will create an electric field and a time varying electric field will create a magnetic field. At one particular point the E field will be varying in a sinusoidal manner with respect to time and so will the B field. It is the time variation of one field that produces the other. Now as Bob travels alongside the laser beam what does he observe? He observes this sinusoidal E field in space as well as a sinusoidal B field. However, this is a static situation. E & B are constant in time. But from Maxwell s equations this can t be! There is no source whatsoever for these fields, no electric charges and no time varying fields. There are three ways to try to amend this: [1] If Maxwell s Equations are valid in Bob s frame then these fields do not exist! Then comparing with Alice s view there is a serious problem. Alice sees a light wave, an entity containing energy but Bob does not! The energy of the universe in Bob and Alice s frame is different. How can this be!? Also if Bob slows down ever so slightly then he would observe a light wave. [2] If Maxwell s equations are not valid in Bob s frame, then why aren t they? Bob s frame is a valid IRF just like Alice s. It is highly unlikely that the laws of physics would change depending upon how you move. For we said before that the laws of mechanics are all the same in all IRFs. [3] A third possibility is that Maxwell s equations are valid and that he observes a light wave travelling at c. This will require some work and tearing down of some long held philosophical viewpoints. Option Galilean Rel. Maxwell s Eqns Bob sees Alice sees Problem?. [1] valid valid no light light at c Violation of energy. For which IRF is v = c? [2] valid modify light at v light at c Light appears at different speeds. Michelson-Morley Expt says no. Again, for which IRF is v = c? [3] modify valid light at c light at c None, except our intuitions will be taken for a wild ride. This third option is what Einstein chose. To allow the first possibility, first Einstein extended the relativity principle of mechanics (that Newton s equations are valid in all IRFs) to all of physics (including Maxwell s etc.). 9

10 Principle of Relativity: ALL physical laws hold in all IRFs. (Not just Newton s) [PSR] This implies that in Bob s frame Maxwell s equations are valid. Again we are discussing all reference frames which are moving with respect to one another at constant velocity. Maxwell s equations state that light travels at 3 x 10 8 m/s, so it must in all frames by this principle. So what does Bob see? By the principle, Maxwell s equations are valid in all IRFs and thus he sees a light wave traveling at c = 3 x 10 8 m/s, done. Well, with this postulate we have explained away the problem of the gedankenexperiment. But what does this mean for the rest of physics. In the forthcoming lectures we will explore what the consequences are. The principle above is the central element of Einstein s paper On the Electrodynamics of Moving Bodies (1905) where special relativity is introduced. The principle has an immediate consequence mentioned just previously, here we will call it a corollary of the principle of relativity* Corollary: Constancy of the speed of light [c], The speed of light is constant and equal to c = 3 x 10 8 m/s in all IRFs. *The principle is sometimes stated as two different postulates (the two statements above) however; the constancy of the speed of light is a consequence of the principle of relativity. Lecture 1 additional: Time Dilation The above postulates were put forward by Einstein to resolve some discrepancies between Maxwell s Theory and classical mechanics. (To quote Einstein, The phenomena of the electromagnetic induction forced me to postulate the (special) relativity principle (1919)). What are the consequences of these postulates? This is what we turn to now. To begin, we will loosen some tenets of physics. Classically we had an absolute time; the universe operated as if by one universal clock. In everything you ve learned thus far, time passed at the same rate for all observers in IRFs. In addition, classically there is assumed to be an absolute space; all observers agree on the distance between the Earth and the Sun. In what will follow, these assumptions will be discarded. Only the implications of the two postulates will be assumed. The results of these explorations will seem bizarre but they have all been proved to be experimentally true. Space and time are not as simple as one would naively think. What is interesting to note is that Maxwell s Theory is already compliant with special relativity (this may not seem to surprising as it comes from the first postulate put forward). It will be seen that classical mechanics will need to modified. Before discussing the geometry of space and time simple derivations of the implications of the new principle will be presented. The following argument is presented in almost every introductory discussion of relativity. To see that our naïve view of an absolute time is incorrect, let us imagine a simple gedankenexperiment. Imagine Alice and Bob each have a special clock, a light clock. The light clock is a simple device which measures time by measuring the time difference between bounces of a light pulse between two mirrors. There is an emitter which emits a pulse of light, travels a distance D, and then bounces off a mirror and returns to a detector (see figure). This is a very simple clock, based on fundamental physical principles (the speed of light). A pendulum clock, for example, is a far more complicated device in terms of fundamental physical laws. Alice s clock has D set so that her clock ticks every nanosecond (10-9 s). Bob s clock is set up in the same manner (D = 0.15 m) and ticks every ns as well. The clocks are calibrated in the same frame so that they tick at the same rate in the same frame. Now we set Alice and Bob in motion with respect to each other. As they are both traveling at constant velocity, they can both be considered to be within an IRF. We can not say which is at rest and which is moving, only that they are moving with respect to each other. In Alice s reference frame she sees Bob traveling by her at speed v. In Bob s reference frame he sees Alice pass by at speed v in the opposite direction. Let s see what Alice observes when she looks at Bob s clock. 10

11 The pulse of light travels in a diagonal direction compared to her own light clock. The distance the light travels is greater than the distance the pulse travels in her own clock. If the distance for Alice s clock is D, the distance she observes for Bob s clock is 2 D 2 + L 2. The distance L is just one-half the distance Bob s clock travels in one round trip of his light pulse. According to Alice, the time it takes for Bob s light pulse to reach the mirror and back to his detector is, t = 2 D2 + L 2 c This result stems from the second postulate Alice observes Bob s light pulse traveling at c. (Classically, the light pulse would NOT be traveling at c). Bob observes the time for his own clock to be, t 0 = 2D c = 1ns We see that Alice observes the time between bounces to be greater than what Bob observes. So Alice observes Bob s clock to run slower than hers! Time is not absolute. The observed time depends on the relative motion of observers. This is a radical change in our philosophy of time. (You may be asking, well this is one special clock, if we used my ultra-precise, ultra-expensive, wristwatch then I should not observe a difference. To convince you that this effect goes beyond just this one type of clock, think about the machinations of your watch. If it is a mechanical watch, then all of the contact forces are actually electrical forces, which we have explained are mediated by photons travelling at the speed of light. This speed is the same in all frames and we get the same result. No matter which technology your watch employs, the forces within it are mediated either by electrical, magnetic, or gravitational forces. Hence, the above results will apply. (We have not yet talked about the speed of mediation of gravitational forces, this will be discussed when we get to the general theory. In short, all forces travel at, or less than, the speed of light.)) We can take the two equations for what Alice and Bob observe to determine how much slower Alice observes Bob s clock to run, t = 2 D2 + L 2 c, L = v t 2, = 4D2 c 2 + v2 t 2 c 2 but t 0 = 2D c t = t v2 t 2 t = t 0 1 v 2 c 2. c 2 t 2 (1 v2 c ) = t Since v < c, this relation tells us how much slower Alice observes Bob s clock to run as compared to what Bob observes his own clock to run. N.B.: Alice observes her clock to tick every nanosecond. Bob observes his clock to tick every nanosecond. However, Alice observes Bob s clock to run every nsec 1 v 2 slower. c 2 N.B.: Since there is no preferred IRF (there is no universal frame of absolute rest) Bob observes Alice s clock to run slow by the same amount. 11

12 To summarize; The two postulates of special relativity imply that time is not absolute, the time dilation results we have just seen bear this out. Clocks moving relative to an observer (all at constant velocity) are measured by that observer to run more slowly as compared to clocks at rest with respect to the observer. Appendix 1.A Galilean Transformations and the Galilean Principle of Relativity Galileo developed the Principle of Relativity (what we will call the Galilean or Newtonian Principle of Relativity to distinguish it from Einstein s) which states that: Two observers moving relative to one another at a uniform rate must formulate the laws of nature in exactly the same way. No observer can distinguish between absolute rest and absolute motion by appealing to any law of nature; hence, there is no such thing as absolute motion, but only motion in relation to an observer. The laws of nature in Galileo s and Newton s sense are the mechanical laws of natures (Newton s laws). The relation between observations in two such reference frames moving with respect to each other is given by the Galilean transformations. Consider two reference frames from the perspective of the first, S (with coordinates x,y,z,t). Frame S is at rest and frame S (with coordinates x,y,z,t that are collinear with S) moves uniformly in the x direction with speed v. If the origins coincide at time t = t = 0, the relations between the measurements are, In frame S In frame S x = x + vt y = y z = z t = t x = x - vt y = y z = z t = t If an object is observed to be moving with velocity V in frame S, the observed velocity in frame S is, V x = v + V x V y = V y V z = V z If an object is observed to be accelerating in frame S the observer in S will observe the same acceleration. 12