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.

IB Question:

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:

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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.

E6 Galaxies and Expanding Universe

E6.1 A galaxy is a cluster of stars. A galactic cluster is a cluster of galaxies. Galactic clusters grouped together is called a super galactic cluster.

E6.2 As the universe is expanding there spectra are red shifted. Galaxies that are the furthest away are red shifted the most.

E6.3 

Heinneman HL Physics, Chris Hamper

E6.4 Hubble's law is that the distance of celestial bodies is proportional to the recession velocity.

Heinneman HL Physics, Chris Hamper

The gradient of this line is called Hubble's constant. The value of this constant changes depending on the number of data points we obtain and plot.

Heinneman HL Physics, Chris Hamper

E6.6 The recession velocity is calculated by observing the red shift and using the equation above to obtain the recession velocity. The distance is calculated by using the three methods (Stellar parallax, Spectroscopic parallax and Cepheid variable method). Using this, we can plot the data points on the graph to obtain Hubble's Constant. Units: Recession Velocity in Kms^-1 and Distance in Mpc^-1

E6.8

Hubble's equation...

Hubble's constant is about 72kms^-1Mpc^-1

E6.5 The data points are scattered about the line of best fit which means that we have random errors in our measurement. Also the gravitational attraction slows down the recession speed of the galaxies but we are assuming that the recession speed is constant.

 We don't have data points near the origin for the graph because the red shift is so small that the errors in the apparatus become significant.

E6.7 Hubble's constant's units are kms^-1Mpc^-1 which is ditancetime^-1distance^‑1 which is time^-1 Therefore the inverse of Hubble’s constant is the age of the universe. 

Heinneman HL Physics, Chris Hamper

But we are assuming that the velocity is constant but we know that the universe is slowing down. Therefore the speed we measure today is higher than what it actually is which means that the time measured as the age of the universe is an upper limit.

E5 Stellar Processes and Evolution

E5.1 Stars are formed when huge clouds of dust and gas are compressed. This cannot occur on its own as the gravitational attraction between them is not strong enough. Star formation is initiated by either a supernova explosion or by two dust clouds' collision. As gas clouds are pulled closer together, the gravitational potential energy converts to kinetic energy causing an increase in the temperature. Also as the particles get closer together the force between them increases causing a rise in the pressure of the dust cloud. The cloud therefore collapses into an increasingly hotter body.

The centre of the dense core rapidly contracts leading to high temperatures and pressure. This is called a protostar but cannot be seen as it is surrounded by a cloud of dust.

After 100000 years the radiation from the star blows away the dust and its mass stabilizes. It now forms a pre-main sequence star. The abundance of hydrogen in the universe suggests that the star is made mainly of hydrogen. When fusion reaction starts taking place, the radiation pressure eventually equals the gravitational pressure to form a body of stable mass. This is now a main sequence star like the sun.


E5.2 5.3Eventually when the hydrogen runs out, the gravitational pressure starts increasing causing the star to collapse. As this occurs the temperature and pressure of the star increases and it becomes possible to fuse helium into higher elements. However, these fusion reactions release an even greater amount of energy which causes the start to expand. When the star expands, the outer layers become cooler. This therefore forms a red giant star. The process of fusing higher and higher elements ensues till be reach iron which has the greatest binding energy per nucleon. After this fusing iron to form higher elements would be endothermic and the star cannot continue to shine. At this point, when the star has run out of nuclear fuel, it starts collapsing due to gravitational force. From here it may either turn into a neutron star or, if it is big enough, into a black hole.

E5.4
The initial mass of the dust cloud determines the size of the star that forms from it which helps us determine where different stars join the main sequence.

Hinemann HL Physics, Chris Hamper

E5.5

E4- Cosmology

E4.1Newton's model of the universe: Newton believed that the universe is infinite in space and time, uniform and static (otherwise it would collapse under its own gravity). This would means that the universe has an infinite number of stars spreading out to infinity.

E4.2Olber's Paradox: If Newton's model was true then this would imply that the night sky ins infinitely bright as there are an infinite number of stars in the sky.

Suppose stars are distributed in an infinite number of thin shells. If each has a Luminosity of L, related to the apparent brightness (b) and distance d, this would imply that the apparent brightness is inversely proportional to the square of the distance.

If there was a thin shell of stars with thickness T at distance d then the volume of the shell would be:
d

Heinneman HL Physics, Chris Hamper

E4.3 On observing galaxies through the Hubble telescope we have observed a red shift in a majority of their spectra which suggests that they are moving away from us, supporting the notion that the universe is not static but is expanding.

E4.4: If galaxies are moving away from each other, logically, this must mean that at some point in the past they were closer together. Extending this idea, we know that once, the entire universe must have been concentrated in a single spot called the singularity. We believe that about 13 billion years ago, the universe must have begun with an explosion called the Big Bang. Space and Time began following the Big Bang.

E4.5 4.6 4.7: Cosmic Microwave Background radiation (CMB radiation) was observed by Penzias and Wilson when they received excess noise from their radio. This is an echo of the Big Bang still echoing in the universe. Penzias and Wilson found that the intensity of the radiation that they received corresponded to that of waves in the microwave region. Substituting this in Wien's displacement law equation they found the average temperature of the universe to be 2.7 K. 

The Big Bang model resolves Olber's paradox as if the galaxies are moving away from each other and are red shifted then this means that we won't be able to see them. This is why the night sky appears dark.

E4.8

Heinemann HL Physics - Chris Hamper

Open Universe: This will continue to expand. Gravity will slow it down but not stop the expansion

Closed Universe: This will eventually collapse back on itself; this is called the big crunch and the reverse of the big bang will occer.

Flat universe:This will

E4.9 Critical density  is the mass per unit volume  that would create a flat universe.

E4.10 If the actual density of the universe is lower than critical density then this would create an open universe, but is the actual density of the universe is higher than the critical density then this would create a closed universe

E4.11 

Dark matter is matter that is not visible as it emits little or no radiation.

WIMPs  are Weakly Interacting Massive Particles. These are hypothetical particles serving as one possible solution to the dark matter problem. They are held together by weak forces and gravity. As they do not interact through electromagnetic radiation it is not possible to detect them and as they do not interact with strong nuclear force, they do not create atomic nuclei. (http://en.wikipedia.org/wiki/WIMPs)

MACHOs are massive astrophysical compact halo objects. This is another hypothesis as to what dark matter is which claims that dark matter is made of the same substance as 'ordinary matter' but just happens to emit little or no radiation making them invisible. (http://news.bbc.co.uk/2/hi/8508662.stm)

It is difficult to determine the density of the universe because we cannot detect dark matter as it does not emit electromagnetic radiation; therefore we know neither its exact volume nor its mass. Knowing the density of the universe in important because this will allow us to determine the fate of the universe; whether it is open, closed or flat.

E12. Current scientific evidence suggests that we have an open universe.

E13. International projects include the study of Drake's equation which consists of a series of variables that, when multiplied together, give the probability of us finding extra terrestrial life. 

Other links:

http://insciences.org/article.php?article_id=8795- Finding dark energy

http://insciences.org/article.php?article_id=6839- Black holes

E14. 

Arguments for astrophysics:

• Understanding the nature of the universe and attempting to answer questions like why we are here and whether or not there is intelligent life other than us
• Develops technology which may improve the quality of life in the future
• To colonise new planets if, in the future, the earth becomes inhabitable
• To understand the fate of the universe and what our future might be

Arguments against astrophysics:

• Money could be used for economic development instead
• Money could be used to conduct more pragmatic research  like medical or energy sources
• Is the information really worth the cost?