PHY 102: Lecture History 12.2 Events and Inertial Reference Frames 12.3 Postulates of Special Relativity 12.4 Relativity of Time: Time Dilation 12.5 Relativity of Length: Length Contraction 12.6 Relativistic Momentum 12.7 Equivalence of Mass and Energy

PHY 102: Lecture 12 Relativity 12.1 History

Galileo Dialogue Concerning the Two Chief World Systems –any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments

PHY 102: Lecture 12 Relativity 12.2 Events and Inertial Reference Frames

Event An event, such as the launching of the space shuttle, is a physical “happening” that occurs at a certain place and time

Observers Two observers are watching the lift-off One standing on the earth One in an airplane that is flying at a constant velocity relative to the earth

Reference Frame – 1 Each observer uses a reference frame This consists of a set of x, y, z axes (coordinate system) and a clock Coordinate systems are used to determine where the event occurs Clocks are used to specify when

Reference Frame – 2 Each observer is at rest relative to his own reference frame The earth-based observer and the airborne observer are moving relative to each other

Inertial Reference Frame An inertial reference frame is one in which Newton’s law of inertia is valid If the net force acting on a body is zero, the body either remains at rest or moves at a constant velocity Acceleration of such a body is zero when measured in an inertial reference frame Earth and airplane are approximately inertial reference frames

PHY 102: Lecture 12 Relativity 12.3 Postulates of Special Relativity

Postulates of Special Relativity Einstein built his theory of special relativity on two fundamental assumptions or postulates about the way nature works 1.The Relativity Postulate. The laws of physics are the same in every inertial reference frame 2.Speed-of-Light Postulate. The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to each other

Relativity Postulate Since the laws of physics are the same in all inertial frames, there is no experiment that can distinguish between an inertial frame that is at rest and one that is moving at a constant velocity It is not possible to single out one particular inertial reference frame at “absolute rest” According to Einstein, only the relative velocity between objects, not their absolute velocities, can be measured and is physically meaningful

Speed-of-Light Postulate - 1 Speed-of-light postulate defies common sense A person standing on the bed of a truck that is moving at a constant speed of 15 m/s relative to the ground You are standing on the ground and the person on the truck shines a flashlight at you

Speed-of-Light Postulate - 2 The person on the truck observes the speed of light to be c You might think that the ground observer measure the speed of light to be c+15 m/s However, the Speed-of-Light Postulate and experiment tells us that the ground observer would also see the speed of light to be c

Speed-of-Light Postulate - 3 Scientists before Einstein assumed that light required a medium through which to propagate This hypothetical medium was called the luminiferous ether and was assumed to fill all of space It was believed that light traveled at the speed c only when measure with respect to the ether According to this view, an observer moving relative to the ether would measure a speed for light that was slower or faster than c, depending on whether the observer moved with or against the light, respectively

Speed-of-Light Postulate - 4 During the years 1883 – 1887, the American scientists A. A. Michelson and E. W. Morely carried out a series of famous experiments The results were not consistent with the ether theory Their results indicated that the speed of light is indeed the same in all inertial reference frames and does not depend on the motion of the observer These experiments, and others, led eventually to the demise of the ether theory and the acceptance of the theory of special relativity

PHY 102: Lecture 12 Relativity 12.4 Relativity of Time: Time Dilation

Time Dilation - 1 Common experience indicates that time passes just as quickly for a person standing on the ground as it does for an astronaut in a spacecraft Special relativity reveals that the person on the round measures times passing more slowly for the astronaut than for himself

Light Clock Short pulse of light is emitted by light source Reflects from a mirror Then strikes a detector that is next to source Each time a pulse reaches the detector, a “tick” registers on the chart recorder Then another short pulse of light is emitted, and the cycle repeats The time interval between successive “ticks” is marked by a beginning event (the firing of the light source) and an ending event (the pulse striking the detector) The source and detector are so close to each other that the two events can be considered to occur at the same location

Time Dilation - 2 Suppose two identical clocks are built One is kept on earth, and the other is placed aboard a spacecraft that travels at a constant velocity relative to the earth The astronaut is at rest with respect to the clock on the spacecraft and, therefore, sees the light pulse move along the up/down path

Time Dilation - 3 According to the astronaut, the time interval  t 0 required for the light to follow this path is the distance 2D divided by the speed of light c  t 0 = 2D/c To the astronauts,  t 0 is the time interval between the “ticks” of the spacecraft clock That is, the time interval between the beginning and ending events of the clock.

Time Dilation - 4 An earth-based observer does not measure  t 0 as the time interval between these two events Since the spacecraft is moving, the earth-based observer sees the light pulse follow the diagonal path This path is longer than the up/down path seen by the astronaut. But light travels at the same speed c for both observers, in accord with the speed of light postulate The earth-based observer measures a time interval  t between the two events that is greater than the time interval  t 0 measured by the astronaut The earth-based observer using her own earth-based clock to measure the performance of the astronaut’s clock, finds that the astronaut’s clock runs slowly This result of special relativity is known as time dilation To dilate means to expand, and the time interval  t is “expanded” relative to  t 0

Time Dilation - 5 The time interval  t that the earth-based observer measures can be determined as follows While the light pulse travels from the source to the detector, the spacecraft moves a distance 2L = v  t to the right v is the speed of the space craft relative to the earth The light pulse travels a total diagonal distance of 2s during the time interval  t

Time Dilation - 6 Applying the Pythagorean theorem, we find that The distance 2s is also equal to the speed of light times the time interval  t 2s = c  t

Time Dilation - 7 Therefore, Squaring this result and solving for  t gives

Time Dilation - 8 But 2D/c =  t 0, the time interval between successive “ticks” of the spacecraft’s clock as measured by the astronaut With is substitution, the equation for  t can be expressed as Time dilation:

Time Dilation - 9  t 0 = proper time interval, which is the interval between two events as measured by an observer who is at rest with respect to the events, and who views them as occurring at the same place  t = dilated time interval, which is the interval measured by an observer who is in motion with respect to the events and who views them as occurring at different places v = relative speed between the two observers c = speed of light in a vacuum For a speed v that is less than c, the term is less than 1, and the dilated time interval  t is greater than  t 0.

Problem 12.1 A spacecraft is moving past the earth at a constant speed v that is 0.92 times the speed of light This written as v = 0.92c The astronaut measures the time interval between successive “ticks” of the spacecraft clock to be  t 0 = 1.0 s What is the time interval  t that an earth observer measures between “ticks” of the astronaut’s clock?

Problem The Global Positioning System (GPS) uses highly accurate and stable atomic clocks on board each of 24 satellites orbiting the earth at speeds of 4000 m/s These clocks make it possible to measure the time it takes for electromagnetic waves to travel from a satellite to a ground-based GPS receive From the speed of light and the times measured for signals from three or more of the satellites, it is possible to locate the position of the receiver The stability of the clicks must be better than one part in to ensure the positional accuracy demanded of the GPS

Problem Calculate the difference between the dilated time interval and proper time interval as a fraction of the proper time interval and compare the result to the stability of the GPS clocks This result is approximately 1000 times greater than the GPS-clock stability of one part in If not taken into account, time dilation would cause an error in the measured position of the earth-based GPS receiver roughly equivalent to that caused by a thousand-fold degradation in the stability of the atomic clocks In 1 second, the earth-based clock is in error by 1.1 x s In 1000 s the error is 1.1x10 13 Therefore, the GPS system fails in 1000 s or about 20 minutes

Proper Time Interval - 1 Both the astronaut and the person standing on the earth are measuring the time interval between a beginning event (the firing of the light source) and an ending event (the light pulse striking the detector) For the astronaut, who is at rest with respect to the light clock, the two events occur at the same location Being at rest with respect to a clock is the usual or “proper” situation, so the time interval  t 0 measured by the astronaut is called the proper time interval

Proper Time Interval - 2 The proper time interval  t 0 between two events is the time interval measured by an observer who is at rest relative to the events and sees them at the same location in space On the other hand, the earth-based observer does not see the two events occurring at the same location in space, since the spacecraft is in motion The time interval  t that the earth-based observer measures is, therefore, not a proper time interval in the sense that we have defined it

Proper Time Interval - 3 To understand the situations involving time dilation, it is essential to distinguish between  t 0 and  t It is helpful if one first identifies the two events that define the time interval These may be something other than the firing of a light source and the light pulse striking a detector Then determine the reference frame in which the two evens occur at the same place An observer at rest in this reference frame measures the proper time  t 0.

Problem Alpha Centauri is 4.3 light-years away This means that, as measured by a person on earth, it would take light 4.3 years to reach this star If a rocket leaves for Alpha Centauri and travels at a speed of v = 0.95c relative to the earth, by how much will the passengers have aged, according to their own clock, when they reach their destination? Assume that the earth and Alpha Centauri are stationary with respect to one another

Problem The two events are the departure from earth and arrival at Alpha Centauri At departure, earth is just outside the spaceship Upon arrival at the destination Alpha Centauri is just outside Therefore, relative to the passengers, the two events occur at the same place Thus, the passengers measure the proper time interval  t 0 on their clock

Problem For a person left behind on earth, the events occur at different places, so such a person measures the dilated time interval  t To find  t we note that the time to travel a given distance is inversely proportional to the speed Since it takes 4.3 years to traverse the distance between earth and Alpha Centauri at the speed of light, it would take even longer at the slower seed of v =0.95c Thus, a person on earth measures the dilated time interval to be  t = (4.3 years)/0.95 = 4.5 years This value can be used with the time-dilation equation to find the proper time interval  t 0

Problem Thus, the people aboard the rocket will have aged only 1.4 years when they reach Alpha Centauri, and not the 4.5 years an earthbound observer has calculated

Problem Verification of Time Dilation The average lifetime of a muon at rest is 2.2 x s. A muon created in the upper atmosphere, thousands of meters above sea level, travels toward the earth at a speed of v = 0.998c Find, on the average, (a) how long a muon lives according to an observer on earth, and (b) how far the muon travels before disintegrating.

Problem (a) The observer on earth measures a dilated lifetime given by (b) The distance traveled by the muon before it disintegrates is x = v  t = (0.998)(3.00 x 10 8 m/s)(35 x s) = 1.0 x 10 4 m Thus, the dilated, or extended, lifetime provides sufficient time for the muon to reach the surface of the earth If its lifetime were only 2.2 x s, a muon would travel only 660 m before disintegrating and could never reach the earth

PHY 102: Lecture 12 Relativity 12.5 Relativity of Length: Length Contraction

Length Contraction - 1 Because of time dilation, observers moving at a constant velocity relative to each other measure different time intervals between two events Both the earth-based observer and the rocket passenger agree that the relative speed between the rocket and earth is v = 0.95c

Length Contraction - 2

Length Contraction - 3 Since speed is distance divided by time and the time is different for the two observers, it follows that the distances must also be different, if the relative speed is to be the same for both individuals Thus, the earth observer determines the distance to Alpha Centauri to be L 0 = v  t = (0.95c)(4.5 years) = 4.32 light-years

Length Contraction - 4 On the other hand, a passenger aboard the rocket finds the distance is only L = v  t 0 = (0.95c)(1.4 years) = 1.3 light years The passenger, measuring the shorter time, also measures the shorter distance This shortening of the distance between two points is one example of a phenomenon known as length contraction

Length Contraction - 5 The relation between the distances measured by two observers in relative motion at a constant velocity can be obtained with the aid of the diagram Part a of the drawing shows the situation from the point of view of the earth-based observer. This person measures the time of the trip to be  t, the distance to be L 0, and the relative speed of the rocket to be v = L 0 /  t

Length Contraction - 6 Part b of the drawing presents the point of view of the passenger, for whom the rocket is at rest, and the earth and Alpha Centauri appear to move by at a speed v The speed to be v = L/  t 0 Since the relative speed computed by the passenger equals that computed by the earth-based observer, it follows that v = L/  t 0 = L 0 /  t Using this result and the time-dilation equation, we obtain the following relation between L and L 0 :

Length Contraction - 7 Length contraction The length L 0 is called the proper length It is the length between two points as measured by an observer at rest with respect to them

Length Contraction - 8 Since v is less than c, the term is less than 1, and L is less than L 0 It is important to note that this length contraction occurs only along the direction of motion Those dimensions that are perpendicular to the motion are not shortened

Problem 12.5 An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter of the spacecraft to be 82 and 21 m, respectively The spacecraft moves with a constant speed of v = 0.95c relative to the earth. What are the dimensions of the spacecraft, as measured by an observer on earth? The length L of the spacecraft, as measured by the observer on earth, is Both astronaut and the observer on earth measure the same value for diameter of spacecraft: Diameter = 21 m

Proper Time and Length - 1 When dealing with relativistic effects we need to distinguish carefully between the criteria for the proper time interval and the proper length The proper time interval  t 0 between two events is the time interval measured by an observer who is at rest relative to the events and sees them occurring at the same place All other moving inertial observers will measure a larger value for this time interval The proper length L 0 of an object is the length measured by an observer who is at rest with respect to the object All other moving inertial observes will measure a shorter value for this length The observer who measures the proper time interval may not be the same one who measures the proper length

Proper Time and Length - 2 The astronaut measures the proper time interval  t 0 for the trip between earth and Alpha Centauri The earth-based observer measures the proper length L 0 for the trip.

Proper Time and Length - 3 It should be emphasized that the word “proper” in the phrases proper time and proper length does not mean that these quantities are the correct or preferred quantities in any absolute sense If this were so, the observer measuring these quantities would be using a preferred reference frame for making the measurement, a situation that is prohibited by the relativity postulate According to this postulate, there is no preferred inertial reference frame When two observers are moving relative to each other at a constant velocity, each measures the other person’s clock to run more slowly than his own, and each measures the person’s length, along that person’s motion, to be contracted

PHY 102: Lecture 12 Relativity 12.6 Relativistic Momentum

Relativistic Momentum - 1 When two or more objects interact, the principle of conservation of linear momentum applies if the system of objects is isolated This principle states that the total linear momentum of an isolated system remains constant at all times The conservation of linear momentum is a law of physics, and, in accord with the relativity postulate, is valid in all inertial reference frames That is, when the total linear momentum is conserved in one inertial reference frame, it is conserved in all inertial reference frames Relativistic momentum is not p = mv

Relativistic Momentum - 2 Magnitude of the relativistic momentum The magnitudes of the relativistic and nonrelativistic momentum differ by the factor, Since this factor is always less than 1 and occurs in the denominator, the relativistic momentum is always larger than the nonrelativistic momentum

PHY 102: Lecture 12 Relativity 12.7 Equivalence of Mass and Energy

Total Energy of Object Mass and energy are equivalent Consider an object of mass m traveling at a speed v Einstein showed that the total energy E of the moving object is related to its mass and speed Total energy of a object

Rest Energy Rest energy of an object The rest energy represents the energy equivalent of the mass of an object at rest When an object is accelerated from rest to a speed v, the object acquires kinetic energy in addition to its rest energy The total energy E is the sum of the rest energy E 0 and the kinetic energy KE, or E = E 0 + KE Therefore, the kinetic energy is the difference between the object’s total energy and its rest energy

Kinetic Energy For low values of v this becomes KE = ½ mv 2

Problem Sun radiates electromagnetic energy at the rate of x W What is the change in the sun’s mass during each second that it is radiating energy?

Energy Any change in the rest energy of a system causes a change in the mass of the system It does not matter whether the change in energy is due to a change in electromagnetic energy, potential energy, thermal energy, or so on Although any change in energy gives rise to a change in mass, in most instances the change in mass is too small to be detected Examples are (1) electron / positron annihilation, and (2) electron / positron pair production by a gamma ray