lecture4 0706

Looking at a Moving Clock: 

Looking at a Moving Clock Consider a flashing clock (i.e. one that emits a flash of light every time it ticks). When at rest it ticks every t seconds. You are watching this flashing clock through a telescope as it moves away from you at a velocity v. According to you it ticks every seconds. What time elapses between your observations of successive ticks? Answer: you observe a tick once every

Slide3: 

Source in rest frame that emits pulses with a frequency f0 = N pulses/s : This is known as the Relativistic Doppler Effect

Manifestations of the Doppler Effect: 

Manifestations of the Doppler Effect Red shift of distant galaxies (Edwin Hubble, Hubble’s law, Hubble telescope). Astronomy: Venus rotates in the opposite direction of Earth. Radar Laser Cooling

The speed of any material object as measured by any inertial observer cannot exceed c: 

The speed of any material object as measured by any inertial observer cannot exceed c u’ = 4 mph v = 8 mph u = 12 mph u = u’+v A-station v = 0.9c u’ = 0.9c B C A launches B at 0.9c, then B launches C at 0.9c, non-relativistically one expects that C leaves A at 1.8c. Answer: 0.994c

Slide6: 

Consider C to be equipped with a flashing clock that takes a time t between ticks in its rest frame. B will see a flash every These flashes go past B and are observed by A, who can regarded at coming either from C’s clock (proper time: t) or, equally well, as coming from a flasher of B’s (proper time given by above) A will see a flash every as if coming from C. A will see a flash every as if coming from B. A-station B C v = 0.9c u’ = 0.9c u = ?

Slide7: 

Let’s do the algebra … When u’ and v << c, then u  u’ +v When either u’ or v = c, then u = c, giving us the principle of the constancy of c. When u’ and v < c, u must also < c. It is impossible to get an object moving faster than light by a succession of pushes, each of which increases its velocity by any amount < c.

Why nothing can go faster than light: 

Why nothing can go faster than light We just saw that the usual way of achieving high velocities, by continually increasing the velocity by small amounts, can never result in a speed greater than light. But it does not rule out the possibility that something might be moving with v > c that it had attained in some unknown way (or that it always possessed). Suppose a ship traveled between Earth and Sirius at a speed u > c. In system in which Earth is at rest trip starts at x = 0 and t = 0, and Sirius is at a distance L. So in this system the ship arrives at Sirius at x = L and t = L/u. Consider a second observer moving along the line from Earth to Sirius at speed v. He also assigns departure at x’ = 0 and t’ = 0. Then the arrival If u > c, then c2/u < c, and v large enough to make (c2/u -v) < 0, then ship arrives before it leaves. Nothing can go faster than light.

Slide9: 

Consider a stick with a mirror on the right end. At a given moment a photon and a particle moving with velocity u’ leave the left end, moving to the right along the stick. The photon reaches the mirror first, is reflected, and moving back to the left, encounters the particle still moving to the right at a point whose distance from the left is a fraction x of the total length of the stick. Show, using only the two principles, that an observer moving to the left with velocity v with respect to the stick will find the velocity of the particle to be . Use the fact that x is an invariant quantity.

2.6: Addition of Velocities: 

2.6: Addition of Velocities Taking differentials of the Lorentz transformation, relative velocities may be calculated:

So that…: 

So that… defining velocities as: ux = dx/dt, uy = dy/dt, u’x = dx’/dt’, etc. it is easily shown that: With similar relations for uy and uz:

The Lorentz Velocity Transformations: 

The Lorentz Velocity Transformations In addition to the previous relations, the Lorentz velocity transformations for u’x, u’y , and u’z can be obtained by switching primed and unprimed and changing v to –v:

Slide13: 

Three identical radio transmitters A, B, and C, each transmitting at the frequency f0 in its own rest frame, are in motion as shown. (a) What is the frequency of B’s signals as received by C? (b) What is the frequency of A’s signals as received by C? A B C -v +v

Slide14: 

Consider the three block labeled 1, 2, and 3 (see figure). Block 1 slides at speed c/2 relative to the ground; block 2 slides at speed c/2 relative to block 1; and block 3 slides at speed c/2 relative to block 2. (a) Find the speed of block 2 relative to the ground. (b) Find the speed of block 3 relative to the ground. 1 2 3

Spacetime Diagrams: 

Spacetime Diagrams When describing events in relativity, it is convenient to represent events on a spacetime diagram. In this diagram one spatial coordinate x, to specify position, is used and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length. Spacetime diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams. Paths in Minkowski spacetime are called worldlines.

Spacetime Diagram: 

Spacetime Diagram

Worldlines in Spacetime Diagram: 

Worldlines in Spacetime Diagram

Worldlines in Spacetime Diagram: 

Worldlines in Spacetime Diagram

Particular Worldlines: 

Particular Worldlines

Worldlines and Time: 

Worldlines and Time

Moving Clocks: 

Moving Clocks

Inertial Frames in Relative Motion: 

Inertial Frames in Relative Motion

Inertial Frames in Relative Motion: 

Inertial Frames in Relative Motion

The Light Cone: 

The Light Cone

Spacetime Interval: 

Spacetime Interval Since all observers “see” the same speed of light, then all observers, regardless of their velocities, must see spherical wave fronts. s2 = x2 – c2t2 = (x’)2 – c2 (t’)2 = (s’)2

Spacetime Invariants: 

Spacetime Invariants If we consider two events, we can determine the quantity Δs2 between the two events, and we find that it is invariant in any inertial frame. The quantity Δs is known as the spacetime interval between two events.

Spacetime Invariants: 

Spacetime Invariants There are three possibilities for the invariant quantity Δs2: Δs2 = 0: Δx2 = c2 Δt2, and the two events can be connected only by a light signal. The events are said to have a lightlike separation. Δs2 > 0: Δx2 > c2 Δt2, and no signal can travel fast enough to connect the two events. The events are not causally connected and are said to have a spacelike separation. Δs2 < 0: Δx2 < c2 Δt2, and the two events can be causally connected. The interval is said to be timelike.

Slide30: 

The Twin Paradox Homer and Ulysses are identical twins. Ulysses travels at a constant high speed to a star and returns to Earth while Homer remains at home. Homer’s clock: it takes Ulysses 5 y to get to the star and 5 y to come back. Thus Homer is 10 years old when Ulysses comes back. If v= 0.8c, then the time interval recorded by Ulysses is 6 years (10/). Ulysses on the other hand should expect Homer to have aged only 3.6 years (6/).

Slide31: 

Once each year, each twin will send a light signal to the other.

Slide33: 

The Pole and Barn Paradox A runner carries a 10 m pole (rest length) toward a 5 m long barn. The runner carrying the pole at speed v enters the barn and at some instant the farmer sees the pole completely contained in the barn: a 10-m pole into a 5-m barn!! The minimum speed v: For the runner, the pole is 2.5 m long. How can a 2.5 m barn enclose a 10 m pole ?????

Slide34: 

From the farmers’s point of view

Slide35: 

From the runner’s point of view