The Theory of Special Relativity

The Theory of Special Relativity

Learning Objectives • Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd postulate that the speed of light is the same in all reference frames: 1. Time dilation: moving clocks run (tick) slower. Proper Time. • 2. Length contraction: moving rods contract. Proper Length. • 3. Loss of simultaneity. • What causes the Doppler shift for light? Time dilation, together with geometrical effects. • Velocity transformation between inertial reference frames.

Time Dilation • Suppose that a strobe light located at rest in frame S´ flashes every ∆t´ seconds. If one flash is emitted at time t1´, the next flash is emitted at time t1´+ ∆t´. What is the interval between flashes as measured in frame S? • Using Eq. (4.24) (inverse Lorentz transformation for time) • we get • and with x1´ = x2´ we find • Which is larger, ∆t or ∆t´ ? We are not concerned with geometrical time differences at this stage, which will be included when we consider the Doppler shift of light.

Time Dilation • Replace the strobe light by a clock located at rest in frame S´. The second hand of this clock ticks every 1s as measured in frame S´. What is the time interval between ticks for the second hand of this clock as measured in frame S? • Following the previous derivation, we find • so that, ∆t > ∆t´. Thus, an observer in frame S measures an interval of >1s between ticks for the second hand of the clock at rest in frame S´. If all clocks in the S and S´ reference frames were synchronized when the origins of both reference frames coincided, thereafter according to an observer in frame S the clocks in frame S´ are running more slowly than the clocks in frame S. This effect is known as time dilation, whereby ”moving clocks run (tick) slower.” Clock

Time Dilation • Although it may not seem like it, once again all we are doing is applying an additional postulate to Newtonian physics, that is Einstein’s 2nd postulate. • Consider a “light clock” consisting of a light pulse that bounces between two mirrors. Say it takes light 1s to travel from one mirror to the other and back as measured by an observer at rest with respect to this clock. Everytime the light pulse makes a round trip, the second hand of this clock moves by one tick. • If the light clock is in uniform relative motion, we see the light pulse having to travel a larger distance and hence taking longer (i.e., >1s) to travel from one mirror to the other and back. That is, the moving light clock appears to tick more slowly (run slower) than the stationary (at rest with respect to us) light clock. (Note that this result must hold no matter how the clock is oriented, as time can only run at one rate in a given reference frame.) Stationary light clock Light clock in uniform relative motion

Time Dilation and Proper Time • Let us return to the derivation of the time interval between ticks for the second hand of a clock at rest in frame S´ as measured in frame S: • Let us call ∆t´ = ∆trest as the clock is at rest in frame S´. Let us cal ∆t= ∆tmoving as the clock is moving in frame S. Then • the effect of time dilation on a moving clock. • The time interval between two events is therefore measured differently by different observers in uniform relative motion. Which clock measures a shorter time interval, a clock at rest or moving with respect to the events? Clock

Length Contraction • Suppose that a rod lies at rest along the x´-axis of frame S´. Let the left end of the rod be at x1´ and the right end at x2´, so that the length of the rod as measured in frame S´ is L´ = x2´ − x1´. What is the length of the rod as measured in frame S? • From Eq. (4.16) • we find that • and with t1 = t2 (naturally both ends of the rod are measured at the same time in frame S) • Which is shorter, L or L´ ?

Length Contraction and Proper Length • Once again, we are simply applying an additional postulate to Newtonian physics, that is Einstein’s 2nd postulate. • Consider the light clock described before, but now with one clock oriented orthogonal to and the other clock parallel to the direction of motion of the S´ frame. They must both tick at the same rate as measured by an observer in the S frame (as time can only run at one rate in a given reference frame), although suffering from the effect of time dilation. • According to the observer in the S frame, if there is no length contraction, the light pulse of the orthogonally-oriented clock makes a round trip in a shorter time interval than the light pulse of the parallel-oriented clock. (Imagine that the S´ frame is stationary and the S frame is being carried to the left by a river: the cross-stream swimmer makes a round trip faster than the upstream-downstream swimmer.) This cannot happen as both clocks tick at the same rate, implying that the parallel-oriented clock must suffer length contraction.

Length Contraction and Proper Length • Let us return to the derivation for the length of a rod in frame S´ as measured by an observer in frame S where the rod aligned in the direction of motion: • Let us call L´ = Lrest as the rod is at rest in frame S´. Let us cal L = Lmoving as the rod is moving in frame S. Then • Lengths (distances) are therefore measured differently by two observers in relative motion. Who measures the longer length, an observer at rest or moving with respect to the object? • Note that only lengths (distances) parallel to the direction of relative motion are affected by length contraction. Lengths (distances) perpendicular to the direction of relative motion remain unchanged (c.f. Eqs. 4.17-4.18).

Time Dilation and Length Contraction • So far, we seem to have treated time dilation and length contraction separately, but these are not independent effects. The effects of time dilation and length contraction are seen together: an observer in the S frame will measure a clock at rest in the S´ frame to run slower and be narrower in the direction of motion than an identical clock at rest in the S frame, as shown in the figure below. • Similarly, an observer in the S´ frame will measure the clock at rest in the S frame to run slower and be narrower in the direction of motion than an identical clock at rest in the S´ frame. This is not contradictory, but simply a consequence of the constancy of the speed of light in all reference frames. • Not only is time dilation and length contraction not independent, they are complementary as illustrated in the following example. Assignment question Clock

Time Dilation and Length Contraction • Muon survival as inferred by an observer at rest with respect to Mt. Washington. muons 0.9952c Time for muon to travel from top to bottom of Mt. Washington 563 muons hr-1 Muon half-life Mt. Washington Time dilation! 1907 m What is wrong with this calculation? muons 0.9952c 408 muons hr-1

Time Dilation and Length Contraction

Time Dilation and Length Contraction • Muon survival as inferred by an observer travelling along with a muon. muons 0.9952c Time for muon to travel from top to bottom of Mt. Washington 563 muons hr-1 Muon half-life Mt. Washington 1907 m What is wrong with this calculation? muons 0.9952c 408 muons hr-1

Time Dilation and Length Contraction

Learning Objectives • Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd postulate that the speed of light is the same in all reference frames? 1. Time dilation: moving clocks run (tick) slower. Proper Time. • 2. Length contraction: moving rods contract. Proper Length. • 3. Loss of simultaneity. • What causes the Doppler shift for light? Time dilation, together with geometrical effects. • Velocity transformation between inertial reference frames.

Loss of Simultaneity • Suppose in frame S two flashbulbs go off at the same time t but at different x-coordinates x1 and x2. Would the two flashbulbs also go off at the same time in frame S´? y’ x1' x2' u x1 x2 x’ O’ Our laboratories in the S and S´ frames have rulers and clocks (i.e., observers) everywhere to measure the local coordinates of events. z’

Loss of Simultaneity • Sometimes, this kind of question is framed in a deliberately confusing (wrong) way: suppose an observer in frame S measures two flashbulbs go off at the same time t but at different x-coordinates x1 and x2. Would an observer in frame S´ also see the two flashbulbs go off at the same time? y’ y’ ⇔ x1' x2' x1' x2' u u x1 x2 x1 x2 x’ x’ O’ O’ Our laboratories in the S and S´ frames have rulers and clocks (i.e., observers) everywhere to measure the local coordinates of events. For an observer in frame S to measure two flashbulbs going off at the same time tbut at different x-coordinates x1 and x2, this observer must be located midway between the two flashbulbs. z’ z’

Loss of Simultaneity • Sometimes, this kind of question is framed in a deliberately confusing (wrong) way: suppose an observer in frame S measures two flashbulbs go off at the same time t but at different x-coordinates x1 and x2. Would an observer in frame S´ also see the two flashbulbs go off at the same time? y’ y’ ⇔ x1' x2' x1' x2' u u x1 x2 x1 x2 x’ x’ O’ O’ Our laboratories in the S and S´ frames have rulers and clocks (i.e., observers) everywhere to measure the local coordinates of events. Where is the observer in frame S´ located? We are not concerned with the effects of geometrical delay. To be sensible, this question requires the observer in frame S´ to be located everywhere; i.e., observers at every location in space. z’ z’

Loss of Simultaneity • Suppose in frame S two flashbulbs go off at the same time t but at different x-coordinates x1 and x2. Would the two flashbulbs also go off at the same time in frame S´? • Using Eq. (4.19) we find that • The observer in the S´ frame therefore does not see the two flashbulbs going off at the same time, but instead that the flashbulb at x1´ goes off after the flashbulb at x2´! This result is simply a consequence of applying Einstein’s 2nd postulate, which implies the downfall of universal simultaneity. y’ x1' x2' u x1 x2 ≠ 0 (> 0 if x2 > x1) x’ O’ z’

Loss of Simultaneity • Both the observer on the platform and us are not moving relative to each other. Both the observer and us see the two flashbulbs going off at the same time. • The platform is now moving to the right relative to us. What if we applied the principle that the speed of light is the same in all reference frames? • We see the platform observer moving away from flashbulb 1, and so light from flashbulb 1 has to travel a greater distance to reach the platform observer. We see the platform observer moving towards flashbulb 2, and so light from flashbulb 2 travels a shorter distance to reach the platform observer. For light traveling at the same speed from both flashbulbs (Einstein’s 2nd postulate) to reach the platform observer at the same time, flashbulb 1 must go off before flashbulb 2 as we see it.

Loss of Simultaneity • The concept of the loss of simultaneity is counterintuitive, and problems are often posed in such a way so as to create a paradox. In all cases, the paradox results from an incorrect application of this concept. • For example, suppose two cars collide according to a stationary pedestrian. Because of the loss of simultaneity, does this mean that the two cars do not collide according to a person who drives by?

Loss of Simultaneity • Here is the correct way to pose a problem that illustrates the loss of simultaneity. • Suppose two car accidents occur at different locations along the same road, and at the same time according to two pedestrians having synchronized watches at the same locations. Would the two car accidents occur at the same time according to two drivers having synchronized watches who drive by at the same speed at both locations?

Doppler Shift for Sound Waves • As a source of sound waves moves through air, the wavelength is compressed in the forward direction and expanded in the backward direction. In Newtonian physics, this change in wavelength is purely a geometrical effect caused by the motion of the source relative to the observer, and is perceived by the observer as a change in the tone of a source depending on its speed and whether it is moving towards or away from us.

Doppler Shift for Sound Waves • A useful way to understand the Doppler effect for sound, and which (as we shall see) provides a useful comparison with the Doppler effect for light, is the following. • Suppose I throw one ball at you every second. Each ball represents a wavefrontof sound. If I stand still, will you receive less than one, one, or more than one ball every second?

Doppler Shift for Sound Waves • A useful way to understand the Doppler effect for sound, and which (as we shall see) provides a useful comparison with the Doppler effect for light, is the following. • Suppose I throw one ball at you every second. Each ball represents a wavefront of sound. If I move towards you, will you receive less than one, one, or more than one ball every second according to Newtonian physics?

Doppler Shift for Sound Waves • A useful way to understand the Doppler effect for sound, and which (as we shall see) provides a useful comparison with the Doppler effect for light, is the following. • Suppose I throw one ball at you every second. Each ball represents a wavefront of sound. If I move away from you, will you receive less than one, one, or more than one ball every second according to Newtonian physics?

Doppler Shift for Sound Waves • In 1842, the Austrian physicist Christian Doppler deduced that the difference between the wavelength obs observed for a moving source of sound and the wavelength rest of the same source of sound at rest is related to the (radial) velocity of the source such that where vs is the speed of sound, and vr the speed of the source relative to the observer (positive when moving apart). (Note that this equation does not depend on whether the sound-carrying medium, air, is moving or not.) Sound waves

Doppler Shift for Light Waves • As a source of light waves moves, the wavelength is compressed in the forward direction and expanded in the backward direction. In Newtonian physics, this change in wavelength is purely a geometrical effect caused by the motion of the source relative to the observer, and is perceived by the observer as a change in the color of a source depending on its speed and whether it is moving towards or away from us. Light waves • In 1842, the Austrian physicist Christian Doppler deduced that the difference between the wavelength obs observed for a moving source of light and the wavelength rest of the same source of light at rest is related to the (radial) velocity of the source such that c where c is the speed of light, and vr the speed of the source relative to the observer (positive when moving apart).

Doppler Shift for Light Waves • The previous equation for the Doppler effect of light is wrong! To see why, consider the following. • Suppose, according to my watch, I throw one ball at you every second. If I am walking towards you, will you see me throw less than one, one, or more than one ball every second according to Special Relativity?

Doppler Shift for Light Waves • The previous equation for the Doppler effect of light is wrong! To see why, consider the following. • Suppose, according to my watch, I throw one ball at you every second. If I am walking towards you, will you see me throw less than one, one, or more than one ball every second according to Special Relativity? According to you, my watch runs more slowly than yours (time dilation). So, according to you, I throw less than one ball every second. • Suppose, according to you, I throw one ball every 1.5 s. If I am walking towards you, will you receive less than one, one, or more than one ball every 1.5 s? More than one ball every 1.5 s. This is purely a geometrical effect, just like in the Doppler effect of sound. • Suppose, according to you, I throw one ball every 1.5 s. If I am walking away from you, will you receive less than one, one, or more than one ball every 1.5 s? Less than one ball every 1.5 s. This is purely a geometrical effect, just like in the Doppler effect of sound. • Doppler effect for light therefore involves time dilation and geometrical effects.

Doppler Shift for Light Waves • The previous equation for the Doppler effect of light is wrong! To see why, consider the following. • Suppose, according to my watch, I throw one ball at you every second. If I am walking towards you, will you see me throw less than one, one, or more than one ball every second according to Special Relativity? According to you, my watch runs more slowly than yours (time dilation). So, according to you, I throw less than one ball every second. • If I am moving perpendicular to you, will you receive less than one, one, or more than one ball every second? Less than one ball every second, reflecting the effect of time dilation. There is no geometrical effect involved here. This situation is known as the transverse Doppler effect, for which there is no parallel in Newtonian physics.

Doppler Shift for Light • Consider a distant light source that emits a light signal at time trest,1 and another light signal at time trest,2 as measured by a clock at rest relative to the source. If this light source is moving relative to an observer at a velocity u, then the time between receiving the light signals at the observer’s location will depend on: - the effect of time dilation, as the interval between light signals is different as measured by the observer and by the clock at rest relative to the source - the purely geometric effect of a time difference between when the two signals reach the observer • Note: we assume that the light source is sufficiently far away that the signals travel along parallel paths to the observer. This assumption is made to simplify the expression for the geometrical time difference. If this assumption does not hold, all that required is an appropriate (more complicated) expression for the geometrical time difference.

Doppler Shift for Light • From Eq. (4.27), we find that the time between signals as measured in the observer’s frame is (due to time dilation) • In this time, the observer determines that the distance to the light source has changed by an amount (due to a purely geometrical effect) Time interval as measured in frame at rest with respect to light source. Time interval as measured in frame where light source is moving.

Doppler Shift for Light • From Eq. (4.27), we find that the time between signals as measured in the observer’s frame is (due to time dilation) • In this time, the observer determines that the distance to the light source has changed by an amount (due to a purely geometrical effect) • Thus, the time interval between the arrival of the two light signals at the observer’s location is Time interval as measured in frame at rest with respect to light source. Time interval as measured in frame where light source is moving. Speed of light is constant irrespective of the relative motion of the light source. Time dilation Geometrical time delay

Doppler Shift for Light • If ∆trest is taken to be the time between emission of light wave crests, then the frequency of the light wave as measured in the frame of the moving source is υrest = 1/∆trest. • If ∆tobs is taken to be the time between arrival of light wave crests at the observer’s location, then the frequency of the light wave as measured by the observer is υobs = 1/∆tobs. • Thus, from Eq. (4.31) • we have • where vr = ucosθ is the radial velocity of the light source. This is the equation for the relativistic Doppler shift. Time dilation Geometrical time delay

Doppler Shift for Light • If the light source is moving directly away from the observer (θ= 0°, u = vr), then Eq. (4.32) • reduces to • In this definition, vr is positive if source is moving radially away from you, and negative if source is moving radially towards you. Time dilation Geometrical time delay

Doppler Shift for Light • Even if the light source is not moving toward or away from the observer, but instead is moving perpendicular to the observer (θ= 90°), the light source is still Doppler shifted. In this case, Eq. (4.32) • reduces to • This effect is called the transverse Doppler effect. What is the transverse Doppler effect due to? Time dilation Geometrical time delay

Redshifts 26 January 2011 • What do astronomers mean by redshift? This galaxy was discovered in the Hubble Ultra Deep Field (HUDF), which is an image of a small region of space in the constellation Fornax that was composited from Hubble Space Telescope data accumulated over a period from September 24, 2003, through to January 16, 2004 (total exposure of 11.6 days over 400 orbits). It is the deepest image of the universe ever taken, looking back approximately 13 billion years (between 400 and 800 million years after the Big Bang), and has been used to search for galaxies that existed at that time. The HUDF image was taken in a section of the sky with a low density of bright stars in the near-field, allowing much better viewing of dimmer, more distant objects. The image contains an estimated 10,000 galaxies.

Redshifts

Redshifts • Because of the expansion of the Universe, galaxies appear to be moving away from each other • Is light from galaxies Doppler shifted to shorter or longer wavelengths because of the expansion of the Universe?

Redshifts • Furthermore, more distant galaxies appear to be moving faster away from: spectral lines in light from more distant galaxies are shifted to longer λ’s.

Redshifts • Astronomers usually express recession velocities as redshifts (z).

Redshifts • When a source of light moves away from us (vr > 0), the frequency of a given spectral line that we measure is shifted to a lower frequency according to Eq. (4.33) • so that υobs < υrest. Equivalently, the wavelength of the spectral line is shifted to a longer wavelength, an effect known as redshift. • When a source of light moves toward us (vr < 0), the frequency of a given spectral line that we measure is shifted to a higher frequency according to Eq. (4.33) so that υobs > υrest. Equivalently, the wavelength of the spectral line is shifted to a shorter wavelength, an effect known as blueshift. Light from galaxies show a Doppler effect due to a combination of the expansion of space and their individual peculiar velocities (if any). The expansion of space causes galaxies to move apart, and hence for galaxies to be carried away from us (and indeed for all galaxies to be carried away from each other). Any motion through space is described by the peculiar velocities of galaxies.

Redshifts • The dark vertical stripes in this figure are spectral absorption lines. The horizontal axis is wavelength, which increases to the right. increasing λ

Redshifts • Astronomers usually express the recession velocities of galaxies as redshifts. • Astronomers define the redshift parameter • The observed wavelength is obtained from Eq. (4.33) • and c = λυ to give

Redshifts • Substituting Eq. (4.35) into Eq. (4.34), we find for the redshift parameter • which expresses the relationship between the redshift and recession velocity of a galaxy.

Redshifts

The Theory of Special Relativity