Today’s Lecture: Special Relativity Galilean Relativity Einstein’s Special Theory of Relativity Length, time, mass, momentum, & energy Extra: Relativity and Space Travel Homework 3 due Thursday, February 21 Reading for Today: Chapter 7 Reading for Next Lecture: Chapter 8

Consider the following “thought” experiment: Imagine you are “floating” freely in a spaceship, because you have no sense of motion you assume you are at rest You look out the window and see your friend Jackie moving by in her spaceship at 90 km/hr. However, she is also freely floating so from her frame of reference you are moving. Both viewpoints must be correct and both agree on the relative speed.

Galilean or Newtonian Relativity We see that motion is relative to your reference frame, however Galilean Relativity assumes some measurements are absolute. It was assumed, that independent of your state of motion (reference frame), that measurements of: Lengths, Time intervals, and Masses will be the same. Distances and times in our everyday life appear absolute. We would expect everyone to agree on the length of an object, the duration of an event, or the mass of an object.

We are familiar with this concept of velocity addition. To the left is an example of a javelin thrower – by running the thrower can increase the speed of the javelin. What about waves ?? Imagine moving toward the source of the water wave. The water wave would appear to be moving faster. Same is true for sound waves. Galilean relativity. How do velocities add ??

Electromagnetic Waves Do all waves needed a medium to propagate (sound travels on air and water waves on water) ?? If so, what is the medium for light ? In the 19 th century, believed that space was permeated by a substance called ether (or aether), which was the medium for light waves. This ether was not detectable. Now, if light added like other waves, one should be able to measure the difference in the speed of light parallel and perpendicular to the motion of the Earth in its orbit about the Sun. (Change about 1 part in 10,000.)

In 1881 Michelson (later in 1887 with Morley) attempted to measure the motion of the Earth relative to the ether by measuring the difference in the speed of light. Found the speed of light was the same – how could this be ? Some suggested that the Earth dragged the ether with it, so that was why the speed was the same. However, then there are problems with the aberration of star light. Some even hypothesized the need for the contraction of space (Lorenz-Fitzgerald contraction) to explain the result. Michelson-Morley Experiment

Einstein Relativity ( Einstein Relativity ( ) Einstein made remarkable discoveries concerning space, time, and gravity. In 1905 published 4 fundamental papers: photoelectric effect, Brownian motion, “On the Electrodynamics of Moving Bodies” (Special Relativity), and the equivalence of mass and energy. The Special Theory of Relativity deals with how space and time are intertwined. Einstein assumed that all motion is relative, but what was the same for all observers were the laws of physics.

Einstein’s Special Relativity Einstein assumed that all motion is relative, however Einstein postulated two absolutes: The laws of nature are the same in all reference frames, independent of their state of motion. The speed of light was the same independent of the motion of the light source or observer. For his postulates to be true, he realized that one had to abandon the idea that lengths, times, and masses are absolute. The consequences concerning lengths and times are quite non-intuitive. However the results of Special Relativity are well tested and appear sound. What is “special” about Special Relativity is that it does not deal with accelerated reference frames or gravity.

Thought Experiment at Low Speeds: 90 km/hr 180 km/hr Imagine that you throw a ball while standing on your spacecraft moving at 90 km/hr. In Jackie’s view point, the ball is moving at 180 km/hr = km/hr. Example of Galilean Relativity

What if light obeyed Galilean relativity: Cars A and B collide in an intersection. If light behaved like other waves, then the light from car A would reach you before the light from car B. If you are nearby, you may scarcely notice the difference. However, viewed from 350 pc away, you will see car A enter the intersection and and leave and hour before car B entered the intersection. No collision ? But car A got dented ? This is a serious paradox !!! Either different frames of reference have different laws of physics or the speed of light must be absolute.

A more tangible example is a distant binary star. Imagine a low mass star (0.1 M ⊙ ) and a massive star (20 M ⊙ ) in circular orbits about their center of mass. The orbital period is 1 year and the low mass star has an orbit size of 2.7 AU. The speed of the low mass star is about 80 km s -1. Imagine viewing the system in the orbit plane: If light was additive, when the low mass star is moving away, the light it produces would have a velocity relative to an outside observer of c – 80 km s -1. Half a year later, when it is moving toward us, the light would have a velocity of c + 80 km s -1. Viewed from ~ 300 pc away, one would see the low mass star on both sides of the massive star simultaneously. A serious problem with the laws of physics if light was simple additive.

The speed of light must be absolute. The correct understanding of light addition is the following:

Consequences of Einstein’s Relativity: At slow speeds …. Imagine bouncing a ball off the ceiling of the train car. For the person in the train, the ball goes up and comes directly down. However, as viewed outside of the train, the ball appears to be going faster, since it also has the forward motion of the train.

Jackie is moving past you at high velocity. She shines a light upward, reflecting it off a mirror, both you and her measure the time elapsed. In your viewpoint, the light follows a longer path than in Jackie’s frame of reference. The speed of light must be the same in both reference frames, but since we see light follow a longer path, what is going on ???. Consider light: Speed same for all observers

I We compare times measured and they do not agree - we conclude, that Jackie’s time must be running slower. Of course Jackie sees just the opposite effect and believes your time is running slower.  Times are not the same - time dilation (or time contraction) !!

Lorentz Factor: We see light take the longer slanted path, length is c  t (t is elapsed time) Jackie sees light take a shorter path, the length is c  t’, where t’ is the elapsed time in her reference frame. The distance Jackie travels, in your reference frame, while the light goes from the floor to ceiling is v  t, where v is the velocity of Jackie relative to you. Time dilation is then (Pythagorean formula): (ct) 2 = (vt) 2 + (ct’) 2 Solving for time, t = t', where is called the Lorentz factor:

Lorentz Contraction: The Lorentz factor () is small for velocities small relative to c, but becomes very large as velocity approaches c. v = 0.1 c, = v = 0.9 c, = 2.3 v = 0.99 c, = 7.1 v = c, = 22.4 v = c, = 70.7 If velocity was only 500 mph (805 kph), then v/c = and the Lorentz factor is only: = !!

Now imagine Jackie shines the light in the direction of her motion as viewed by you. In your view you see Jackie’s time running slow, however since the speed of light is absolute, then what must be going on ? Now Imagine... Light must travel a shorter path. From your point of view, the spacecraft appears to shrink in the direction of motion.  length contraction !!

Thus, in your reference frame, Jackie’s clock appears to move slower, and her spacecraft is contracted in the direction of her motion. It is shrunk by the same Lorentz factor as time has contracted. Of course Jackie thinks the same is true of you.

Correct Velocity Addition: From the Lorentz transformation, can derive the correct velocity addition. v = (v 1 + v 2 )/(1 + v 1 v 2 /c 2 ) Unlike Galilean relativity, the velocities always sum to a value less than the speed of light. Note that no matter how fast objects are moving, their relative velocity will always be < c. Thus, nothing can move faster than the speed of light.

Force, Acceleration, and Mass: Newton’s 2 nd law states that: F = m a. However, if the velocity is close to the speed of light, then only a very small acceleration is possible. Since the force you exerted on Jackie’s spacecraft did not create the acceleration expected, how is this possible ???? Before push After push 0.98 c 0.99 c

Force, Acceleration, and Mass: The explanation in that the mass that goes into Newton’s 2 nd law [F = d(mv)/dt] has increased. We call this the inertial mass or relativistic mass. Therefore, the inertial mass of Jackie’s spacecraft has appeared to you to have increased. The inertial mass is not an invariant to motion.  mass increases The inertial mass is given by: m o, where m o is called the rest mass and is larger than what would be measured if Jackie's spacecraft was at rest relative to us.

Summary: From these thought experiments, we can conclude that many properties, such as time, length, and inertial mass, are affected by relative motion. Only, when the relative motion is close to the speed of light, does this produce a noticeable difference. From our viewpoint the following is observed: Time in moving frame appears to be moving slower by Length in moving frame appears contracted by Mass in moving frame appears larger by where Neither time or length are invariant, but there is an invariant in special relativity and that is the quantity: x 2 + y 2 + z 2 - (ct) 2

Our view on Earth 0.99 c Late afternoon you are watching a rerun of Seinfeld. An alien spacecraft flies by at 0.99 c (Lorentz factor about 7) and the aliens peer over your shoulder. The alien’s view of Seinfeld is different, the images are contracted and the show lasts for 3.5 hours !!! The alien’s view as they pass by.

Tests of Relativity: The constancy of the speed of light now measured to one part in Extremely accurate clocks have been flown on airplanes at high speed and compared with clocks remaining at rest. Measure the time dilation and results agrees with the prediction of the Special Theory of Relativity. Sub-atomic particles can approach the speed of light. Accelerators that accelerate electrons to 99% of the speed of light need only be 1 inch long. However, to accelerate electrons to much closer to the speed of light, need accelerator 2 miles long (Stanford Linear Accelerator) due to the increasing inertial or relativistic mass of the electron.

A more everyday example: Old Tube Televisions The electrons being fired from the electron gun in old tube TVs are traveling with a speed of about c. Their relativistic (moving) mass is about 1%-2% higher than their rest mass. If television makers didn’t account for this increase, the picture would come out blurry. Numerous tests of Special Relativity have been performed, all continue to support relativity.

Reconsider Momentum and Energy: The standard Newtonian conservation of momentum and energy do not hold. What is conserved in your reference frame are: Relativistic Momentum: p = m o v Relativistic Energy: E = m o c 2 Consider the relativistic energy: E = m o c 2 = [1 – (v 2 /c 2 )] -1/2 m o c 2 We can use an approximation for (1 – x) -n for small x: (1 – x) -n ~ 1 + nx + n(n+1) x 2 /

Rest Mass Energy Thus for small v 2 /c 2 can use just first two terms: E ~ m o c 2 [1 + ½ (v 2 /c 2 )] or E ~ m o c 2 + ½ m o v 2 Even if an object is not moving it has non-zero energy, this is the rest energy and related to the rest mass by m o c 2. Total energy is the sum of its rest mass energy plus kinetic energy. Einstein concluded mass and energy are equivalent. The famous expression E = m o c 2 is just telling us that if you remove/add energy from a system, then the mass associated with the energy is also removed or added. It does NOT say that mass is converted into energy.

Application of Mass/Energy Equivalence: Example - Formation of a Deuteron ( 2 H) from a proton and neutron: Rest mass of proton = x gm, Rest mass of neutron = x gm, Sum of masses: x gm. Rest mass of deuteron = x gm, Mass difference = x gm Binding energy is x gm x c 2 = 3.5x10 -6 ergs or 2.2 MeV. Would need 2.2 MeV of energy to break apart the deuteron. If the deuteron is formed from a neutron and proton, then energy is released. The energy is: [m d – (m p + m n )] c 2 = 2.2 MeV and the mass decreased by 1 part in 857 – a measurable change.

Example: Chemical Reactions Even in exothermic chemical reactions the rest mass of the reactants are ever so slightly less than the rest mass of the products. The difference in mass is the difference in binding energy of the reactant and the products. For chemical reactions the binding energy is measured in eV not MeV and only a very small change in mass (a part in 10 billion) is expected. Since the nucleus of atoms are tightly bound by the strong force, there is a much potential for energy generation in nuclear reactions. We will see the energy of the Sun is produced by the fusion of atoms.

Momentum of a Photon With a little algebra you can rewrite the relativistic energy in terms of the relativistic momentum: E = (m 2 c 4 + p 2 c 2 ) 1/2 Note that photons with zero rest mass still have momentum: E photon = p photon c or p photon = E photon /c Of course we already know that E photon = hν or hc/λ The momentum of a photon is then: p photon = h/λ It is from this relation that de Broglie got the idea of what should be the wavelength of a particle (λ = h/p).

Space Travel - Extra The speed limit in the Universe poses a problem for space travel to the stars. Remember the nearest star is about 4.3 light years away. Even light takes 4.3 years !!!! Also remember that the fastest spacecraft currently available has a speed of about 60,000 km/hr, only % the speed of light. At this speed, it would take 18,000 years to reach the nearest star !!

However, if we could travel close to the speed of light we do have one trick – Special Relativity. Consider a roundtrip journey to a star at a distance of 70 light years (relatively nearby star). Assume you can travel at 0.99 c (Lorentz factor of about 7). For those remaining on Earth, they will see you travel at near the speed of light for about 70 years, and then take about 70 years to return, elapsed time about 140 years. Special Relativity and Space Travel

From the rest frame of the Earth, the spacecraft will appear to be traveling at 0.99 c and shortened to about one-seventh of its original length. However, from the spacecraft, the Earth and star appear to be moving at 0.99 c and distance between the Earth and star shortened to one-seventh its original distance. The astronauts would complete their journey to the star in 10 years and return in another 10 years, however 140 years would have elapsed on the Earth.