E7 General Relativity

Gravitational Mass: The gravitational mass of an object is the ratio of the gravitational force applied to an object and the strength of the gravitational field.

Inertial Mass: The force applied to an object divided by the acceleration it produces.

These two are identical for all intents and purposes on earth and would only become significant if we accelerate in deep space where the gravitational forces are negligible.

Einstein's Thought Experiment:

Consider two scenarios: If there is an astronaut doing an experiment of dropping a ball in a stationary elevator on earth or in an accelerating elevator. Einstein states that the observations would be identical and the astronaut would not be able to determine whether he is in an accelerating frame of reference or in a gravitational field. This is called the principle of equivalence.

Bending of Light:
Light bends in the presence of bodies with large masses. Evidence: During a solar eclipse in 1919, Arthur Eddington found that stars behind the sun were shifted outwards as shown:

This proves that light from the stars bends due to the presence of a body of large mass like the sun which provides evidence for general relativity.

Gravitational lensing:
Light from distant galaxies behind other galaxies reach the earth because of bending of light. This allows us to see them and is evidence for general relativity.

The distant galaxy appears as a ring around the galaxy in front which is called the Einstein ring.

Time ticks slower in accelerating frames of reference. Consider an beam of light being shone at a wall inside a room that is moving at constant velocity and in a room that is accelerating:

To an observer in the room accelerating, the light appears to be hitting the same spot as he is stationary with respect to the event. But to an observer outside the room, the light appears to be taking a curved path. As the speed of light is constant and the distance that light travels increases according to the observer (curved distances are longer than linear ones), time must be slowing down.

This is why light ticks slower closer to bodies with large masses, as light bends around them.

E6- Relativistic Kinematics

H5 - Evidence to Support Special Relativity

Muon Experiment:

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Evidence:

Muons are produced 10 km above the surface of the earth when cosmic rays from the sun interact with the atmosphere giving them speeds of 0.99c. They have a half life of 2.2 * 10^-6 seconds which means that they should only be able to travel about 1 km before decaying and we should not be able to detect any muons at the surface of the earth. But we do.

gamma factor = 7.1

Conclusions:

As the muons is travelling close to the speed of light, special theory of relativity applies. This means that time is dilated; to an observer on earth, the muon's half life will appear to be 7.1 * 2.2 * 10^-6  seconds. Time dilates. Also, the distance that they have to travel appears to be 10/7.1 km = 1.4 km. The distance contracts. As the time dilates and the distance contracts, the muons appear to survive long enough to reach the surface of the earth, which supports the special theory of relativity.

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Michelson Morley Experiment:

 The aim of this experiment was to test for the presence of ether (which is what scientists thought was the medium through which light travels). 

So for sound, if the wind was moving in the direction of the sound wave, sound would travel faster and if the wind was moving in the opposite direction, then sound should travel slower. They thought the same was true of light and ether. The animation below shows the predicted results of the Michelson Morley experiment:

From:http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm

Method:
In the centre is a half mirror that splits the incident beam of light into two (one reflected and one passed through) that have equal intensities that hit mirrors, converge back at the half mirror and strike the detector.
Depending on  the direction of the ether, one beam was thought to arrive faster than the other.
The experiment was repeated by rotating the apparatus and measuring differences in speed of the return beams.
Interference patterns of the two light beams were used to detect differences in their return time)
But this did not happen, and they both arrived at different times. This proved firstly that ether doesn't exist and that the speed of light is constant.

Pion Decay experiment:
A pion was accelerated close to the speed of light. When it decayed it produced two gamma-ray photons. The speed of these photons was not the sum of the speed of the pion and the photon as Galilean transformations would suggest but was the speed of light. This once again proved the second postulate of the special theory of relativity that the speed of light is constant in all inertial frame's of reference.

H4 - Consequences of Special Relativity

Twin Paradox:
If Nikita was on earth and her twin Atikin was on a very fast journey into space, then from Atikin's perspective Nikita's clock is ticking slower and from Nikita's perspective Atikin's clock is ticking slower. Consequentially they both believe that the other will be younger when they meet on Atikin's return. But they both cannot be younger; this is the twin paradox.

In reality, Atikin will be younger because, for Atikin, to attain that velocity, she must accelerate and before turning to return, she must decelerate. Again, on the return journey she must accelerate to attain that velocity and decelerate on returning to earth. Therefore, she is no longer in an inertial frame of reference. In the accelerating frame of reference, time appears to be ticking slower to an observer in it; therefore, Atikin will be younger on returning to earth.

In order to resolve the twin paradox the general theory of relativity was developed.

Haefele Keating Experiment:

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from: http://www.youtube.com/watch?v=cmPebZA2ZdI

Questions:
from Hienneman:

IB:

H3 Relativistic Kinematics

Light Clock: A beam of light reflected between two parallel mirrors that is used to measure time is called a light clock.


From: http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html

As can be seen, to Jill, her light clock appears to have spanned exactly 10 seconds no matter what her speed. This is because the light clock is stationary with respect to her and the beam of light only has vertical motion not horizontal motion. But if Jack observes Jill's clock, the beam of light from the source is not only moving upwards but also to the right. This means that it has a greater distance to travel than the beam of light approaching the mirror from his light source. As the speed of light is constant, this must mean Jill's clock must be moving slower according to Jack to compensate for the increased distance that  the light beam has to travel.

Proper Time is the time as measured in the frame of reference where the event it occurring. This happens to be the shortest possible time that can be recorded for an event to occur.
Using the above example, both Jack and Jill will measure proper time using their clocks as they are in the frame of reference in which the event is occurring.

Time Dilation Derivation: 

Yeah...you can't really see what's written.
This is what is written:

HL Physics Hienemann, Chris Hamper


Gamma (in the video) is called the Lorentz factor.
The speed of light must be less than c, as otherwise the denominator would be 0 and time would be infinitely long. Therefore, we always measure the time taken for an event to occur in different frame of reference that has relative motion to be longer than the time taken for that event as measured by someone in that frame of reference. Time is said to dilate.

The variation of the Lorentz factor as the velocity of the object varies is shown below:

Graph Data

Time Dilation problems:

Proper Length:Proper length of an object is the length  of an object as measured by an observer in the same frame of reference as the object. This is the longest possible distance of the object that can be measured. As on object approaches the speed of light, its length appears to decrease in the direction of its motion.

If an astronaut is moving between stars close to the speed of light, then the distance between the stars appears to contract which means that the astronaut feels that the distance be travels is shorter and he covers it in a shorter time than what it appears to be from earth. 

From: http://www.youtube.com/watch?v=xvZfx7iwq94&feature=relmfu

Questions on time and distance dilation:

IB Questions:

H2 - Concepts of Special Relativity

Inertial frame of reference is one which is moving at constant speed where Newton's Laws of motion apply. Accelerating frames of reference, like circular motion is a non inertial frame of reference ( there is acceleration towards the centre). Acceleration must be 0 for a frame of reference to be inertial.
Special Relativity: The two postulates for special relativity are

1. The speed of light is constant for all observers in an inertial frame of reference
2. The laws of physics are the same for all inertial observers.

Simultaneity: If two events happen together, they are said to be happening simultaneously.
 Consider

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To a person inside the train moving at constant speed, if both the pulses of light are sent out at the same then the time taken for both beams to hit mirrors on either end and reach back to him is the same. The light pulses return simultaneously. 

But consider a stationary observer on the platform observing this event:

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When the train moves forward, the light travelling towards the left has a shorter distance to travel as the back of the train has moved forward towards the pulse of light. Similarly, the light travelling towards the right has a longer distance to travel as the front of the train is moving forward away from the pulse of light. Speed of light is constant in all inertial frames of reference irrespective of the speed of the frame of reference of the observer. Therefore, to an observer on the platform, the light pulse on the right reaches the mirror after the one on the left.

This event is not simultaneous to an observer on the platform. Two events occurring at different points in space cannot be simultaneous for two different frames of references.

From: http://www.youtube.com/watch?v=AZ6N85lNgHY

H1: Introduction to Relativity

H1: Frame of reference: It is a system of coordinates that allows the position of various objects to be defined and measurements to be made.



From: http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html#intro_to_SR

If object A is moving due east at 20 ms^-1 and object B is moving towards it due west at 15 ms^-1 then

  V_{a rel B} means the velocity of A relative to B i.e. B is the observer and A is the object. 

V_{a rel B} = V_a - V_b = 20 – (-15) = 35 ms^‑1. 

This is an example of a Galilean transformation where time is assumed to be independent of the observer.

Consider:

As they are moving towards each other their relative velocities with respect to each other are twice their original velocities by Galilean transformation = 0.9800c + 0.9800c = 1.960c. This is faster than the speed of light which is not physically possible; therefore at speeds greater than 1c, Galilean transformations break down and consequentially the Theory of Relativity was 'invented.'

But in our everyday life we do not deal with speeds close to the speed of light; therefore the Galilean transformations hold.

The photon moving in the same direction as the pion (photon R) will have a speed of 1c (speed of light/photon)+ 0.900c which is 1.90c. 

The photon moving in the opposite direction as the pion (photon Q) will have velocity 1c - 0.900c = 0.100c.