1. Today’s Lecture:Relativity Einstein’s Special Theory of Relativity Length, Time, Mass, Momentum, & Energy Principle of Equivalence Einstein’s Gravity Homework 3 due Today Homework 4 due Tuesday, March 4 Reading for Today: Chapter 7 and 8 Reading for Next Lecture: Chapter 9.1 – 9.3

2. 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.

3. 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. So how do velocities add ? To the right 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 a water or sound wave. The wave velocity is relative to the media, so it would appear to be moving faster.

4. Electromagnetic Waves Maxwell showed that the electromagnetic waves travel at 300,000 km/s. What is this velocity relative to ? 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. The velocity of light, c, is relative to this ether. 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 (or in other words the motion of the Earth through the ether). Expect a change about 1 part in 10,000.

5. In the 1880s Michelson and 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 starlight. Michelson-Morley Experiment

6. Einstein Relativity ( Einstein Relativity ( 1879-1955) In 1905 Einstein published: “On the Electrodynamics of Moving Bodies” (Special Relativity). Einstein assumed that all motion is relative, but what was the same for all observers were the laws of physics (including Maxwell's laws), thus the speed of light. 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.

7. 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 get dented 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.

8. 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.

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

10. 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 we see light follow a longer path, how can this be explained ???. Consider light: Speed same for all observers

11. 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) !!

12. 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:

13. Lorentz Factor: The Lorentz factor () is close to one for velocities small relative to c, but becomes very large as velocity approaches c. v = 0.1 c, = 1.005 v = 0.9 c, = 2.3 v = 0.99 c, = 7.1 v = 0.999 c, = 22.4 v = 0.9999 c, = 70.7 If velocity was only 500 mph (805 kph), then v/c = 0.00000075 and the Lorentz factor is only: = 1.0000000000003 !!

14. Jackie now shines her light in the direction of her motion as viewed by you. In your view you we have already shown that Jackie’s time running slow. So how does the light traverse her spacecraft in less time ??? Now Imagine... Light must travel a shorter path. From your point of view, her spacecraft appears to shrink in the direction of motion.  length contraction !!

15. 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.

16. 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.

17. Force, Acceleration, and Mass: Newton’s 2 nd law states that: a = F/m. 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

18. Force, Acceleration, and Mass: The explanation in that the mass that goes into Newton’s 2 nd law (a = F/m) 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.

19. Summary: We conclude that 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 going slower by Length in moving frame appears contracted by Mass in moving frame appears larger by where [Neither time or length are invariant, but an invariant in special relativity is the quantity: x 2 + y 2 + z 2 - (ct) 2 ]

20. 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.

21. Tests of Relativity: The constancy of the speed of light now measured to incredible accuracy. Extremely accurate clocks have been flown on airplanes at high speed and compared with clocks remaining at rest. Measured time dilation and agrees with the prediction of the Special Theory of Relativity. In particle accelerators see the affects on the mass of a moving object. Numerous tests of Special Relativity have been performed, all continue to support relativity.

22. Reconsider Momentum and Energy: Different observers will not agree on the energy or momentum of a particle, however what is invariant are the following: 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 /2 +....

23. Equivalence of Mass and 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 sum of its rest mass energy plus kinetic energy. Einstein concluded mass & energy are equivalent. The famous expression E = m o c 2 is telling us that if you add or remove energy from a system, then the mass associated with the energy is also added or removed. Next lecture we will apply this concept.

24. Momentum of a Photon With a little algebra you can rewrite the relativistic energy in terms of the relativistic momentum: E = (m o 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).

25. General Theory of Relativity While Special Relativity deals with space and time for reference frames at rest or in uniform motion, General Relativity deals with accelerating reference frames and gravity. General Relativity was developed through a series of papers published by Einstein between 1907 and 1916. General Relativity changed forever our view of gravity and has a number of important implications for astrophysics.

26. General Theory of Relativity Consider the situation to the left: You fire the rockets on your spacecraft and accelerate away from Jackie. From your viewpoint Jackie is accelerating away from you. Is there a difference ? YES, you feel a force, but Jackie does not. These two views are not equivalent. The principles of General Relativity, like Special Relativity, was developed through a series of thought experiments.

27. However, feeling an acceleration does not necessarily tell who is moving. For instance, standing on the surface of the Earth you feel an acceleration but you do not seem to be moving, while in orbit about the Earth you feel no acceleration but you are clearly moving. Thus, gravity creates accelerations without motions and motions where we feel no sense of acceleration. Einstein reasoned that an observer in free-fall feels no force, so this is the most natural frame. Einstein postulated that the presence of a gravitational field curves space-time, and the motion that results is the straightest possible path in curved space-time. This is a geometrical interpretation of gravity.

28. New View of Gravity What we perceive as gravity arises from the curvature of space-time. The greater the curvature the stronger the effect. The presence of mass curves space-time and the trajectories of particles is determined by this curvature (straightest paths or geodesics)

29. Long before Einstein has a full understanding of this geometrical concept of gravity, he had a realization that was the starting point for general relativity. In 1907, Einstein hit upon an idea that he later claimed was “the happiest thought of my life.” He reasoned that anytime you feel weight, as opposed to feeling weightless, you can equally attribute this to an acceleration or to gravity. This idea is called the equivalence principle: The effects of gravity are equivalent to the effects of acceleration.

30. Equivalence Principle Einstein realized there was no difference between acceleration and gravity – in both cases you feel “weight”. If in a closed room, cannot tell if you are on the Earth’s surface or accelerating in space at 9.8 m s -2. The effects of gravity is equivalent to the effects of acceleration Based on this simple principle, Einstein was able to make some remarkable predictions.

31. Time in Accelerating Frames You and Jackie are at either ends of a spacecraft and you each have clocks that flash every second. At rest or in uniform motion you will both agree that the flashes are 1 second apart. However, if the spacecraft is accelerating upwards your changing reference frame is always carrying you ahead of the point where Jackie’s flashes are emitted. Her flashes will appear slower than yours. Jackie, on the other hand, is accelerating toward your flashes, so to her they appear more frequent. Your time is running faster than hers and you both agree.

32. Equivalence Principle – Time Dilation According to the equivalence principle, acceleration is equivalent to gravity, so if the spacecraft were on Earth, Jackie’s clock would run slower than your clock.  predicted gravitational time dilation.

33. Gravitational Time Dilation: Imagine two clocks, clock 1 is located on the surface of a massive object (M) with a radius R and clock 2 is located very far away. For small time dilations, the ratio of time elapsed is given approximately by: t 2 /t 1 ~ 1 + GM/(R c 2 ) For example, on the surface of the Sun: t 2 /t 1 = 1 + (6.67x10 -8 cm 3 gm -1 s -2 )(2x10 33 gm)/[(7x10 10 cm)(3x10 10 cm s -1 ) 2 ]= 1.000002 Currently, gravitational time dilation can be measured on the Earth for a height difference of only 1 meter !!!!

34. Gravitational Redshift The gravitational redshift was first measured in the lab in 1959, however can readily see the effect in the light from compact objects such as white dwarf stars. Closely related is the gravitational redshift in which light leaving a massive object is redshifted – this can be observed for astronomical sources. Light leaving a massive object is at wavelength (λ 1 ), the wavelength of light seen by a distant observer (λ 2 ), again for small redshifts (see textbook), is given by: λ 2 /λ 1 = 1 + GM/(Rc 2 )

35. Bending of Light In an accelerated frame, light would appear to move in a curved path. Light source At Rest Accelerating Gravity a g Thus, according to the equivalence principle, light is bent by gravity. Needed the full theory published in 1916 to predict the correct deflection.

36. Deflection of Starlight by the Sun An experiment: Sun would be photographed during a total solar eclipse and compared with an image without the Sun. The angle of deflection in radians of light passing within distance b of the center of an object with mass M is: θ(in radians) = 4GM/(b c 2 ) For starlight just passing at the Sun's radius (thus b = R ⊙ ), is deflected by an angle 8.47x10 -6 radians or only 1.74 arcseconds. Measured in the 1919 solar eclipse.

37. Astrophysical Applications: General Relativity has a number of applications to astronomy, and we will be discussing several of these in later lectures. Black Holes Gravitational Lensing Cosmology In the future we may be able to detect gravitational waves – a new way of viewing our Universe.