The Theory of Relativity. What is it? Why do we need it? In science, when a good theory becomes inadequate to describe certain situations, it is replaced by a more general one. This means that the new theory describes more situations than the one it replaces, not that the old one is wrong. Relativity describes the situations when 1.either bodies move at speed close to that of the light; 2.or gravity is very, very strong; 3.or both! –In these cases Newtonian Theory does not work well –Some predictions are wrong; some physical properties do not appear (eg. that energy gravitates) –But the theory predicts black holes just fine –But when speeds are much slower than light and gravity is not too strong, Relativity and Newtonian Theory coincide, are exactly the same!

Assignment Section 5.3 (finish Chapter 5)

How did Einstein think about motion?

Relativity. What is it. Part I? Imagine the empty space, with just yourself and another person in it, and nothing else. You can tell if you move with uniform motion respect to that person (or viceversa). But you cannot say if it is you who is moving, or if it is the other person, or both. In fact, it is not possible to define absolute uniform motion relative to the empty space. But if this is the case, if all observers in uniform motion are “equivalent”. then, all laws of nature must look the same to them –FIRST POSTULATE: The Laws of Physics are the same for all observers, regardless of their motion (as long as not accelerated) otherwise it would be possible to tell one observer from another and define absolute motion.

Relativity. What is it. Part II? In principle the speed of light through space could either be infinite, or finite. It turns out (from direct observations) that it is finite. Then, all observers MUST measure the same speed of light, regardless of their motion. Otherwise, one could use one’s speed relative to light to define absolute motion, which violates the First Postulate. Thus, once we have established as a Law of Physics that the speed of light is finite, then all observers must measure that light moves at the same speed (SECOND POSTULATE). This means that if we have two observers in uniform motion respect to each other (one going one way, the other the opposite way), each shining a flashlight, and both measure the speed of light emitted from the other’s flashlight, they find the same speed, regardless of how fast they move This is against intuition and common sense derived from observations of slow-moving bodies. But it must be true if the FIRST POSTULATE holds

Reference Frames Motion can be defined with respect to a particular frame of reference Ball moves at 10 km/hr in reference frame of plane Ball moves at 910 km/hr in reference frame of someone on ground

Absoluteness of Light Speed Einstein claimed that light should move at exactly c in all reference frames (now experimentally verified) Light moves at exactly speed c (not c km/hr) Light moves at exactly speed c

What’s surprising about the absoluteness of the speed of light? Light moves at exactly speed c (not c km/hr) Light moves at exactly speed c

Thought Experiments Einstein explored the consequences of the absoluteness of light speed using “thought experiments” (Gedankenexperiment) The consequences will be easiest for us to visualize with thought experiments involving spaceships in freely floating reference frames (no gravity or acceleration)

Relativity of Motion at Low Speeds

Relativity of Motion at High Speeds

Light Speed is Absolute c + 0.9c = c !?!

Why can’t we reach the speed of light?

Trying to Catch up to Light Suppose you tried to catch up to your own headlight beams You’d always see them moving away at speed c Anyone else would also see the light moving ahead of you

Special Topic: What if Light Can’t Catch You? Is there a loophole? What if yo are somehow moving away from a distant planet faster than the speed of light? In that case you have no way of detecting that the planet is there. Although there are some phenomena that move faster than light, no information can be communicated faster than the speed of light

What have we learned? How did Einstein think about motion? –Motion must be defined with respect to a reference frame What’s surprising about the absoluteness of the speed of light –Velocities in different reference frames do not add up like we expect them to because the speed of light must be the same for everyone Why can’t we reach the speed of light? –No matter how fast we go, light will always appear to move away from us at speed c

How does relativity affect our view of time and space? Time dilation Both Jackie and you measure the span of the travel time of light in Jackie’s spaceship. Each of you uses own time reference The length of time of the SAME event is different: You measure an event in a frame that moves relative to you, and you measure a longer time span Jackie measure the same even in a the rest-frame, and she measures a shorter time span Let’s se why…

Path of Ball in a Stationary Train Thinking about the motion of a ball on a train will prepare us for the next thought experiment

Path of Ball in a Moving Train Someone outside the train would see the ball travel a longer path in one up-down cycle The faster the train is moving, the longer that path would be

Time Dilation We can perform a thought experiment with a light beam replacing the ball Relative to me, the light beam, moving at c, travels a longer path in a moving frame Time must be passing more slowly there

The Time Dilation Formula Light path in your reference frame Light path in frame of other spaceship

The Time Dilation Time passes more slowly in a moving object by an amount depending on its speed Time almost halts for objects nearing the speed of light Observationally verified!!! Time dilation explains why two working clocks will report different times after different accelerations. For example, ISS astronauts return from missions having aged slightly less than they would have been if they had remained on Earth, and GPS satellites work because they adjust for similar bending of spacetime to coordinate with systems on Earth.

The twin paradox

Simultaneous Events? In your reference frame, red and green lights on other spaceship appear to flash simultaneously

Simultaneous Events? But someone on the other spaceship sees the green light flash first—simultaneity is relative!

Length Contraction Similar thought experiments tell us that an object’s length becomes shorter in its direction of motion

Mass Increase A force applied to a rapidly moving object produces less acceleration than if the object were motionless: –dp = F x dt This effect is equivalent to a mass increase in the moving object

Velocity Addition

Formulas of Special Relativity

Deriving E = mc 2 Kinetic Energy Mass-Energy of object at rest Kinetic Energy is being converted into mass!!!

Do the effects predicted by relativity really occur?

Tests of Relativity First evidence for absoluteness of speed of light came from the Michelson-Morley Experiment performed in 1887 Time dilation happens routinely to subatomic particles the approach the speed of light in accelerators Time dilation has also been verified through precision measurements in airplanes moving at much slower speeds

Tests of Relativity Prediction that E=mc 2 is verified daily in nuclear reactors and in the core of the Sun

Test Relativity for Yourself If speed of light were not absolute, binary stars would not look like two distinct points of light You can verify relativity by simply looking through a telescope at a binary star system

A Paradox of Non-Relativistic Thinking If speed of light were not absolute, you would see the car coming toward you reach the collision point before the car it struck No paradox if light speed is same for everyone

What have we learned? How does relativity affect our view of time and space? –Time slows down for moving objects –Lengths shorten for moving objects –Mass of a moving object increases –Simultaneity of events depends on your perspective Do the effects predicted by relativity really occur? –Relativity has been confirmed by many different experiments

S2.4 Toward a New Common Sense Our goals for learning How can we make sense of relativity? How does special relativity offer us a ticket to the stars?

How can we make sense of relativity?

Making Sense of Relativity According to you, time slows down in a moving spaceship According to someone on that spaceship, your time slows down Who is right? You both are, because time is not absolute but depends on your perspective

Toward a New Common Sense As children we learned that “up” and “down” are relative Relativity tells us that “time” and “space” are relative

How does relativity offer us a ticket to the stars?

A Journey to Vega The distance to Vega is about 25 light-years But if you could travel to Vega at 0.999c, the round trip would seem to take only two years!

A Journey to Vega At that speed, the distance to Vega contracts to only 1 light-year in your reference frame Going even faster would make the trip seem even shorter!

A Journey to Vega However, your twin on Earth would have aged 50 years while you aged only 2 Time and space are relative!

What have we learned? How can we make sense of relativity? –We need abandon our old notions of space and time as absolute and adopt new a new common sense in which time and space depend on your perspective How does special relativity offer us a ticket to the stars? –For someone moving near light speed, distances appear to become shorter because of length contraction

Relativity. What is it. Part III? The implications are far reaching: for example, this means that no matter how much energy you spend to accelerate yourself to a higher and higher speed, even infinite energy, you always find that light has the same speed relative to you and that is faster than you. If you spend infinite amount of energy, and you still cannot reach the speed of light, it is as if your apparent mass is becoming infinite (thus resisting acceleration). But if your apparent mass increases with energy, this means that energy gets transformed into mass (and viceversa). This led Einstein to understand that –E = m x c 2 = (m 0 +  m) x c 2 –m 0 is the mass at rest, the minimum mass;  m is the extra mass coming from the acceleration Mass and energy are different manifestations of the same physical entity, and can transform into each other, given the appropriate conditions. Thus, one can transform m 0 into energy!

Other predictions of Relativity For high-speed motions (v~c) –Time slows down –Length shrinks in the direction of motion For example, if one of two identical twins is accelerated to the nearly the speed of light, kept moving fast for several years, and then slowed back down, he will be younger than the other one, because for him time ticked at a slower pace. All these predictions are verified every day in labs using particle accelerators to accelerate sub-atomic particles to speed close to the speed of light.

How much energy is contained within mass? Consider 1 kg of matter (about 2 lb) Recall that c= 3x10 8 m/s –This means that c 2 = 9x10 16 (m/s) 2 So, the energy associated with 2 lb of matter is: –E = 1 kg x 9x10 16 (m/s) 2 = 9x10 16 Joule This is roughly the energy liberated by the detonation of 20 million tons of TNT (a powerful H-bomb). From only 2 lb of mass… The conversion of matter into energy is the power of the Universe.

The General Theory of Relativity EQUIVALENCE PRINCIPLE: –Observers cannot distinguish locally between inertial forces due to acceleration and gravitational forces due to the presence of a mass To the observer in the room both cases “feel” the same. So, gravity, too, is not an absolute entity, since it is equivalent to an accelerated reference system

How do we do that? How do we get a body of mass M to act as if it was exerting an inertial force (imparting an acceleration) onto another body? Bend the space (and time) around it!

Gravity According to General Relativity Mass tells space-time how to curve, and the curvature of space-time tells mass (another mass) how to accelerate

Gravity and General Relativity The amount of mass and its size (I.e. density) determines gravity, and hence determines the space-time curvature Two bodies with the same mass (Sun’s one) but compressed to different size (density) bends the space-time the same way at large distances, but very differently at small distances

Gravity and General Relativity The amount of mass and its size (I.e. density) determines the space-time curvature Two bodies with the same mass (eg. Sun’s one), but compressed to different size (density) bends the space-time the same way at large distances, but very differently at small ones

Another Thought Experiment Imagine a light source on board a rapidly accelerated space ship: As seen by a “stationary” observer As seen by an observer on board the space ship Light source Time aa a a g

Other Effects of Special Relativity Length contraction: Length scales on a rapidly moving object appear shortened The energy of a body at rest is not 0. Instead, we find E 0 = m c 2 Relativistic aberration: Distortion of angles

Empirical Confirmation of GR. I The precession of Mercury’s orbit is faster than what expected in Newton’s theory, because close to the Sun, gravity is very, very strong. –GR predicts a precession period sec faster than Newton’s theory –Observations measure a period / sec faster than Newton’s theory

Empirical Confirmation of GR But if the space-time is curved, then we expect that the travel path of light is bent as well… –Effect expected to be visible close to the Sun, because there gravity is very, very strong. Except that sun’s glare makes observations impossible. Except during solar eclipses.

Empirical Confirmation of GR Bending of the light (by background galaxies) also observed in proximity of quasars (super-massive black holes) or very, very massive clusters of galaxies (hence, capable of producing very, very strong gravity)

Assigned Reading Chapter 5, all of it