Chapter 24: Special and General Relativity Version 2
The Big Ideas
Einstein believed that the laws of physics do not depend on the how fast you are moving through space: every reference frame sees the same world of physics. In other words, if you are on a moving train and drop a ball or if you are standing on a farm and drop a ball, the physics that describe the motion of that ball will be the same. Einstein realized that the speed of light, c, should depend only on the laws of physics that describe light as electromagnetic radiation. Therefore, Einstein made the bold assertion that light always travels at the same speed, no matter how fast you are moving with respect to the source of light. Consider for a moment how counterintuitive this concept really is. This is the theoretical underpinning of Einstein’s theory of Special Relativity, one of the most successfully predictive theories of physics ever formulated.
The most important consequence of this new understanding is that our intuition that time moves at the same rate for everyone (whether standing still or moving at a fast speed) is WRONG. In fact, the rate at which time passes depends on your speed. Since Einstein’s work in the early part of the century, this fact has been demonstrated many times by experiments in particle accelerators and through the use of atomic clocks aboard fast moving jet airplanes. The effect is only noticeable at extremely fast speeds, thus the normal laws of motion apply in all but the most extreme cases.
Einstein was finally led to believe that the very fabric of space and time must have a more active and influential role in the laws of physics than had previously been believed. Eventually, Einstein became convinced that gravity itself amounted to no more than a curvature in spacetime. This theory is called General Relativity.
Key Concepts
The speed of light will always be measured to be the same (about ) regardless of your motion towards or away from the source of light.
In order for this bizarre fact to be true, we must reconsider what we mean by ‘space,’ ‘time,’ and related concepts, such as the concept of ‘simultaneous’ events. (Events which are seen as simultaneous by one observer might appear to occur at different times to an observer moving with a different velocity. Note that both observers see the same laws of physics, just a different sequence of events.)
Clocks moving towards or away from you run more slowly, and objects moving towards or away from you shrink in length. These are known as Lorentz time dilation and length contraction; both are real, measured properties of the universe we live in.
If matter is compressed highly enough, the curvature of spacetime becomes so intense that a black hole forms. Within a certain distance of a black hole, called an event horizon, nothing can escape the intense curvature, not even light. No events which occur within the horizon can have any effect, ever, on events which occur outside the horizon.
Key Equations
An object moving with speed has a dimensionless speed calculated by dividing the speed by the speed of light (). .
The dimensionless Lorentz “gamma” factor can be calculated from the speed, and tells you how much time dilation or length contraction there is. .
Object Speed (km/sec) Lorentz Factor
Commercial Airplane
Space Shuttle
UFO ?
Electron at the Stanford Linear Accelerator
If you see an object of length moving towards you at a Lorentz gamma factor , it will appear shortened (contracted) in the direction of motion to new length .
If a moving object experiences some event which takes a period of time (say, the amount of time between two heart beats), and the object is moving towards or away from you with Lorentz gamma factor , the period of time measured by you will appear longer.
The radius of the spherical event horizon of a black hole is determined by the mass of the black hole and fundamental constants. A typical black hole radius is about .
The mass of an object moving at relativistic speeds increases by a factor of .
The potential energy of mass is equal to mass times the speed of light squared.
Relativity Example
Question: The muon particle has a half-life of . Most of these particles are produced in the atmosphere, a good - above Earth, yet we see them all the time in our detectors here on Earth. In this problem you will find out how it is possible that these particles make it all the way to Earth with such a short lifetime.
a) Calculate how far muons could travel before half decayed, without using relativity and assuming a speed of (i.e. of the speed of light)
b) Now calculate for this muon.
c) Calculate its ‘relativistic’ half-life.
d) Now calculate the distance before half decayed using relativistic half-life and express it in kilometers. (This has been observed experimentally. This first experimental verification of time dilation was performed by Bruno Rossi at Mt. Evans, Colorado in 1939.)
Answer:
a) To calculate the distance that the muon particle could travel we will use the equation for distance and then plug in the known values to get the answer.
b) To solve for , we must first solve for .
Now we can solve for .
c) To calculate the muon particle's relativistic half-life, we will use the value we calculated in part b) and the equation for determining relativistic half-life.
d) To calculate the distance the muon particle can travel we will use the same distance equation but we will use the new half-life instead of the non-relativistic half-life. Now we will convert this into .
Relativity Problem Set
Suppose you discover a speedy subatomic particle that exists for a nanosecond before disintegrating. This subatomic particle moves at a speed close to the speed of light. Do you think the lifetime of this particle would be longer or shorter than if the particle were at rest?
What would be the Lorentz gamma factor for a space ship traveling at the speed of light c? If you were in this space ship, how wide would the universe look to you?
Suppose your identical twin blasted into space in a space ship and is traveling at a speed of . Your twin performs an experiment which he clocks at minutes. You observe this experiment through a powerful telescope; what duration does the experiment have according to your clock? Now the opposite happens and you do the minute experiment. How long does the traveling twin think the experiment lasted?
An electron is moving to the east at a speed of . What is its dimensionless speed ? What is the Lorentz gamma factor ?
What is the speed of a particle that has a Lorentz gamma factor ?
How fast would you have to drive in your car in order to make the road % shorter through Lorentz contraction?
The muon particle has a half-life of . Most of these particles are produced in the atmosphere, a good above Earth, yet we see them all the time in our detectors here on Earth. In this problem you will find out how it is possible that these particles make it all the way to Earth with such a short lifetime. Calculate how far muons could travel before half decayed, without using relativity and assuming a speed of (i.e. % of the speed of light)
Now calculate , for this muon.
Calculate its 'relativistic' half-life.
Now calculate the distance before half decayed using relativistic half-life and express it in kilometers. (This has been observed experimentally. This first experimental verification of time dilation was performed by Bruno Rossi at Mt. Evans, Colorado in 1939.)
Calculate the radius of the event horizon of a super-massive black hole (SMBH) with a mass times the mass of our Sun. (Unless you have it memorized, you will have to look up the mass of the Sun in .)
If an electron were “really” a black hole, what would the radius of its event horizon be? Is this a measurable size?
An alien spaceship moves past Earth at a speed of with respect to Earth. The alien clock ticks off seconds between two events on the spaceship. What will earthbound observers determine the time interval to be?
In 1987 light reache
d our telescopes from a supernova that occurred in a near-by galaxy light years away. A huge burst of neutrinos preceded the light emission and reached Earth almost two hours ahead of the light. It was calculated that the neutrinos in that journey lost only minutes of their lead time over the light. What was the ratio of the speed of the neutrinos to that of light?
Calculate how much space was Lorentz-contracted form the point of view of the neutrino.
Suppose you could travel in a spaceship at that speed to that galaxy and back. It that were to occur the Earth would be years older. How much would you have aged?
An electron moves in an accelerator at % the speed of light. Calculate the relativistic mass of the electron.
Enterprise crew members notice that a passing Klingon ship moving with respect to them is engaged in weapons testing on board. At the closest point of contact the Klingons are testing two weapons: one is a laser, which in their frame moves at ; the other is a particle gun, which shoots particles at in the Klingon frame. Both weapons are pointed in the same line as the Klingon ship is moving. Answer the following two questions choosing one of the following options: A. B. C. D. E. F. Question 1: What speed, , does the Enterprise measure the laser gun to achieve with respect to the Enterprise?
Question 2: What speed, , does the Enterprise measure the particle gun to achieve with respect to the Enterprise?
How much energy is produced by a kilogram softball?
The isotope of silicon has an atomic mass of . It can go through beta radioactivity, producing with a mass of . Calculate the total energy of the beta particle emitted, assuming the nucleus remains at rest relative to the nucleus after emission.
Another possibility for this isotope is the emission of a gamma ray of energy . How much kinetic energy would the nucleus gain?
What is the frequency and wavelength of the gamma ray?
What is the rebound velocity of the nucleus in the case of gamma ray emission?
Answers to Selected Problems
longer
, the universe would be a dot
.
a. b.
c.
d.
, yes harder to accelerate
a. f b. c
softballs
a. b.
Chapter 25: Radioactivity and Nuclear Physics Version 2
The Big Idea
The nuclei of atoms are affected by three forces: the strong nuclear force, which causes protons and neutrons to bind together, the electric force, which is manifested by repulsion of the protons and tends to rip the nucleus apart, and the weak nuclear force, which causes neutrons to change into protons and vice versa.
The strong force predominates and can cause nuclei of complex atoms with many protons to be stable. The electric force of repulsion is responsible for fission, the breaking apart of nuclei, and therefore also for atom bombs and nuclear power. A form of fission where a helium nucleus is a product, is called alpha radiation. The actions of the weak force give rise to beta radiation. A change in nuclear energy can also give rise to gamma radiation, high energy electromagnetic waves. Particles that emit alpha radiation, beta radiation, and gamma radiation go through the process of radioactive decay, which causes the heating of the molten core of the earth, and has even played a role in the mutations in our evolutionary history. Fission and fusion, the latter occurring when light nuclei combine to form new elements, are accompanied by copious amounts of gamma radiation. These processes often produce radioactive nuclei; in nature these radioactive nuclei were forged in the explosive deaths of ancient stars.
Key Concepts
Atomic symbols like are interpreted in the following way: is the symbol for the element involved. For example, is the symbol for the element uranium. is the atomic number. is the atomic mass number, the total number of nucleons (protons and neutrons). A would be for uranium.
Some of the matter on Earth is unstable and undergoing nuclear decay.
When mass is lost during radioactive decay, the energy released is given by Einsten's famous formula:
Alpha decay is the emission of a helium nucleus and causes the product to have an atomic number lower than the original and an atomic mass number lower than the original.
Beta minus decay is the emission of an electron, causing the product to have an atomic number greater than the original
Beta plus decay is the emission of a positron, causing the product to have an atomic number lower than the original.
When an atomic nucleus decays, it does so by releasing one or more particles. The atom often (but not always) turns into a different element during the decay process. The amount of radiation given off by a certain sample of radioactive material depends on the amount of material, how quickly it decays, and the nature of the decay product. Big, rapidly decaying samples are most dangerous.
The measure of how quickly a nucleus decays is given by the half-life of the nucleus. One half-life is the amount of time it will take for half of the radioactive material to decay.
The type of atom is determined by the atomic number (i.e. the number of protons). The atomic mass of an atom is approximately the number of protons plus the number of neutrons. Typically, the atomic mass listed in a periodic table is an average, weighted by the natural abundances of different isotopes.
The atomic mass number in a nuclear decay process is conserved. This means that you will have the same total atomic mass number on both sides of the equation. Charge is also conserved in a nuclear process.
It is impossible to predict when an individual atom will decay; one can only predict the probability. However, it is possible to predict when a portion of a macroscopic sample will decay extremely accurately because the sample contains a vast number of atoms.
The nuclear process is largely random in direction. Therefore, radiation strength decreases with distance by the inverse square of the distance (the law, which also holds for gravity, electric fields, light intensity, sound intensity, and so on.)
Decay Equations
Nuclear decay is often measured in terms of half lives. During the span of one half life, the amount of a decaying substance decreases by half. Therefore, after half lives, the amount of a substance starting at left is If we need to know the amount left after some time , we first need to see find many half lives transpired (this will be given by , then use the formula above:
If on the other hand, we know how much of a substance is left and would like to find how much time has transpired, we can solve the equation above for (left to reader):
This equation is used in radioactive dating:
Question: The half-life of Pu is years. You have micrograms left, and the sample you are studying started with micrograms. How long has this rock been decaying?
Answer: We will use the equation for time and simply plug in the known values.
Radioactive carbon dating is a technique that allows scientists to determine the era in which a sample of biological material died. A small portion of the carbon we ingest every day is actually the radioactive isotope rather than the usual . Since we ingest carbon every day until we die (we do this by eating plants; the plants do it through photosynthesis), the amount of in us should begin to decrease from the moment we die. By analyzing the ratio of the number of to atoms in dead carbon-based life forms (including cloth made from plants!) and using the technique illustrated above, we can determine how long ago the life form died.
Key Applications
Alpha Decay
Alpha decay is the process in which an isotope releases a helium nucleus ( protons and neutrons, ) and thus decays into an atom with two less protons.
Example:
Beta Decay
Beta decay is the process in which one of the neutrons in an isotope decays, leaving a proton, electron and anti-neutrino. As a result, the nucleus decays into an atom that has the same number of nucleons, with one neutron replaced by a proton. (Beta positive decay is the reverse process, in which a proton decays into a neutron, anti-electron
and neutrino.)
Example:
Gamma Decay
Gamma decay is the process in which an excited atomic nucleus kicks out a photon and releases some of its energy. The makeup of the nucleus doesn’t change, it just loses energy. (It can be useful to think of this as energy of motion – think of a shuddering nucleus that only relaxes after emitting some light.)
Example:
Fission and Fusion
Fission is the process in which an atomic nucleus breaks apart into two less massive nuclei. Energy is released in the process in many forms, heat, gamma rays and the kinetic energy of neutrons. If these neutrons collide with nuclei and induce more fission, then a runaway chain reaction can take place. Fission is responsible for nuclear energy and atom-bomb explosions: the fission of uranium acts as a heat source for the Earth’s molten interior.
Example:
Fusion is the process in which two atomic nuclei fuse together to make a single nucleus. Energy is released in the form of nuclear particles, neutrons, and gamma-rays.
CK-12 People's Physics Book Version 2 Page 23
