139 terms

According to Brian Greene, what do those who subscribe to the occult, astrology, certain religious beliefs and modern physics have in common?

Greene argues that modern physics has repeatedly taught us that "human experience is often a misleading guide to the true nature of reality" and that "lying just beneath the surface of the everyday world is a world we'd hardly recognize" (p. 5). In other words, the reality that we learn about via relativity or quantum mechanics seems very different from the everyday world. Intuitions based on everyday experiences do not apply to the quantum or relativistic realms. Yet these counterintuitive physical regime underlie our everyday experiences. This can be viewed as similar to the way certain belief systems or religions argue that the world of everyday experience is merely a surface, below which there is another layer or aspect of reality. Of course, there are differences too: Unlike various belief systems, the understanding in physics about what "lies beneath the surface" is based on experimental observations of extreme physical circumstances.

In "Chapter 1: Roads to Reality," Greene describes various ways in which modern physics clashes with insights from human experience and intuition. Name at least three examples of this.

There are probably a nearly unlimited number of ways in which modern physics clashes with insights from human experience. Here is a representative set that Green highlights:

• Time and space are not independent or absolute.

• Time and space can warp and curve.

• No matter how much you know about a particular system, you cannot predict exactly what will happen in the future, just the probabilities of various outcomes.

• Things don't assume definite properties until they are observed. Before that, they can be in "mixed" states, simultaneously possessing contradictory features.

• Separated objects are sometimes not physically independent, but can affect each other instantaneously over great distances under the right circumstances.

• There is no difference (in terms of the laws of physics anyway) between going forwards or backwards in time.

• The world may possess more than the familiar three spatial dimensions and one time dimension.

• Time and space are not independent or absolute.

• Time and space can warp and curve.

• No matter how much you know about a particular system, you cannot predict exactly what will happen in the future, just the probabilities of various outcomes.

• Things don't assume definite properties until they are observed. Before that, they can be in "mixed" states, simultaneously possessing contradictory features.

• Separated objects are sometimes not physically independent, but can affect each other instantaneously over great distances under the right circumstances.

• There is no difference (in terms of the laws of physics anyway) between going forwards or backwards in time.

• The world may possess more than the familiar three spatial dimensions and one time dimension.

What force or combination of forces causes this acceleration? (Hint: Why doesn't the person go crashing through the surface of the earth or flying off into space?)

There are two forces that cause this acceleration: The force of gravity, pulling the person towards the center of the earth and the force of the surface of the earth pushing on the person's feet. The second force is necessary because gravity provides far too strong of a pull to supply the needed acceleration. Gravity by itself could provide up to ac = 9.81 m/s2 (this is what happens for an orbiting object). Therefore the outward force of the surface of the earth pushing on the person reduces the net force to just the level needed to keep the person from breaking through the surface of the earth.

Qualitatively, how does the perceived force of gravity on a person vary between someone on the equator compared to someone at the north pole? Explain.

The person at the north pole has zero centripetal acceleration, since they are not really traveling in a circle at all. Therefore, the force of the earth pushing on the person exactly equals the person's weight force. In contrast, because a person on the equator has a centripetal acceleration, which must come from the difference between the weight force and the force of the earth's surface, the earth's surface pushes out a little bit less than the full force of gravity (i.e. the weight force). Since our perception of weight is linked to how hard the surface we're standing on pushes back on us, the person at the north pole would feel slightly heavier (and in fact, a scale would register a slightly larger weight).

Q2

As the skydiver is falling, she is pulled down by the force of gravity, which tends to accelerate her towards the earth, while at the same time the drag force Fd opposes this motion. When she first jumps out of the plane, her speed downward is not very big, and so the drag force is small. As gravity accelerates her downward, her speed increases, and so does the size of the drag force. As the drag force increases, it opposes the force of gravity, decreasing the size of the acceleration. At some point, the drag force will increase to be the same size as the force of gravity. At that point, the skydiver will stop accelerating and hit a constant velocity, known as the terminal velocity.

(b) When the parachute is opened, the b coefficient in the drag increases, and the drag force will be higher than the weight force. This will cause an upward acceleration that will have the effect of reducing the downward velocity. As the velocity decreases, the drag force will decrease, until the drag force once again balances the weight force at a new, lower terminal velocity, hopefully leading a safe landing!

(b) When the parachute is opened, the b coefficient in the drag increases, and the drag force will be higher than the weight force. This will cause an upward acceleration that will have the effect of reducing the downward velocity. As the velocity decreases, the drag force will decrease, until the drag force once again balances the weight force at a new, lower terminal velocity, hopefully leading a safe landing!

Q3

(b) Orbiting occurs when the force of gravity exactly matches the centripetal acceleration for moving in orbit. As the orbital radius increases, the force of gravity gets smaller, while the centripetal acceleration increases (because the speed increases so that the satellite will cover a bigger radius orbit in the same time). The orbital radius is determined by when these two quantities are matched.

Q4

(a) It is reasonable to assume that in a person (or in any other object) the number of electrons is roughly equal to the number of protons. Furthermore, we can get an estimate of the number of protons in a person by assuming that the proton mass is approximately the same as the neutron mass, that the electron mass is negligible, and that for every proton, there is approximately one neutron.

Removing 0.1% = 0.001 of the electrons would result in a net positive charge for each person equal to...

Removing 0.1% = 0.001 of the electrons would result in a net positive charge for each person equal to...

Q7 (see word doc)

The lower ball will win. When the balls start, the lower ball translates more of its energy from potential to kinetic, meaning it moves faster. Then from that point on, the lower ball is always lower than the upper ball, which means it is always moving faster. And if two balls are in a race, and one is always faster, then that one will win. (Note: both balls end the race at the same speed, by conservation of energy, but they don't hit the end at the same time.)

Q9

(b) We can find the wavelength of the sound using fλ = v so that λ = f/v = (340 Hz)/(340 m/s) = 1 m. Therefore we are standing 2 wavelengths in from of one speaker, but 2.5 wavelengths away from the other.

(c) Because the two sound waves are arrive exactly 1/2 a wavelength out of phase, the peak of one wave will align with the trough of the other and you will hear no sound. (It will completely cancel out—also known as completely destructive interference.)

(d) Now you will be an equal distance from both speakers, so the waves will arrive in phase and will interfere constructively, creating a louder sound.

[Note: In a real room, sound bouncing and echoing off the walls and other surfaces will prevent the complete cancelation of sound from the two speakers.]

(c) Because the two sound waves are arrive exactly 1/2 a wavelength out of phase, the peak of one wave will align with the trough of the other and you will hear no sound. (It will completely cancel out—also known as completely destructive interference.)

(d) Now you will be an equal distance from both speakers, so the waves will arrive in phase and will interfere constructively, creating a louder sound.

[Note: In a real room, sound bouncing and echoing off the walls and other surfaces will prevent the complete cancelation of sound from the two speakers.]

Explain in your own words why the principle of Galilean relativity says that the statements "bodies at rest tend to stay at rest" and "bodies in motion tend to stay in motion" are really just the same statement twice.

Because according to Galilean relativity all motion a constant speed is relative, an object which is at rest for one observer will be in motion for another observer. If an object remains at rest for observer A, and observer B moves with a constant velocity with respect to observer A, the the object will remain in constant motion with respect to observer B. In other words, by applying a Galilean coordinate transformation, one can take an object that's at rest (and tending to stay at rest) into an object that is in motion (and tending to stay in motion) for any object.

We generally treat the earth as if it is an inertial (or Galilean) reference frame; however, with all this circular motion, it actually isn't one. And yet, for the most part, Newton's Laws, etc. seem to apply. Why is it safe to ignore the earth's acceleration?

Although these systems are always accelerating, if we look at the size of the acceleration, it's fairly small (see table above). For example, the largest acceleration, that coming from the earth's rotation is only about 0.3% of the acceleration due to gravity (g). Therefore, the effects caused by the earth not being an inertial frame are also small and can (for the most part) be safely ignored.

In Chapter 5 of Relativity, Einstein gives two arguments for why the principle of relativity in the restricted sense should be considered valid. In his first argument, we explains that relativity is built into mechanics (i.e. Newton's Law's, the kinematic equations, etc.) and since this theory does a great job of explaining most of the things that we observe, then it seems unlikely that it's wrong. However, he follows that with a second, more subtle but also more general argument. Please summarize his reasoning in your own words. You may refer to direct quotes from the text, but please follow each quote with your own explanation of what is being said.

Einstein argues that if the principle of relativity weren't valid, then the laws of physics would change depending on the reference frame. Different frames would have different laws and presumably the laws in some frames would be more complicated while in other frames the laws would be simpler. From all the different possible references frames/laws of physics, you should be able to select one that has the least complicated laws of physics. In that case, you'd be justified in calling this reference frame the "best" or "at rest" frame, and all the other reference frames would build on that.

If such a "best" frame exists, the earth is certainly not in it continuously. The motion of the earth is constantly changing as the earth spins on its axis, orbits the sun, and moves with the solar system around the center of the galaxy. Sometimes the earth might be at rest with respect to the "best" frame, but at other times it wouldn't be. So, if the principle of relativity were not true, we should see changes in the laws of physics over time. These changes should be observable to us (at least after careful measurements) and therefore, if the principle of relativity were wrong, we should already know it.

If such a "best" frame exists, the earth is certainly not in it continuously. The motion of the earth is constantly changing as the earth spins on its axis, orbits the sun, and moves with the solar system around the center of the galaxy. Sometimes the earth might be at rest with respect to the "best" frame, but at other times it wouldn't be. So, if the principle of relativity were not true, we should see changes in the laws of physics over time. These changes should be observable to us (at least after careful measurements) and therefore, if the principle of relativity were wrong, we should already know it.

Imagine that you are an international spy. You regain consciousness in a small windowless room. A video screen lights up, and your evil nemesis, Dr. Crims N. Tide, informs you that you are actually sealed in a capsule that is at this moment plunging rapidly down a deep shaft towards the center of the earth. He explains that you feel no effects of this because relativity tells us that all motion is relative and without the ability to compare your motion to another object's, you perceive yourself to be at rest. He informs you that the only way to save yourself from a fiery crash is if you divulge the password to President Obama's twitter account. How should you respond? Explain your answer in terms of physics.

Here you should equate "plunging rapidly down" to mean "falling." If you are in free-fall, you are accelerating, which you can certainly detect without being able to view outside your capsule. In fact, if you are truly in free-fall, you should feel weightless. Dr. Tide's explanation

would only apply if your capsule was moving with a constant speed (e.g. being lowered by a giant winch). If Dr. Tide is willing to go to all the trouble to lower you with a perfectly constant velocity into the center of the earth so that you can't tell what's happening, then he probably deserves to beat you. You should just call his bluff—tell him that your superior knowledge of physics tells you that since you don't feel the tell-tale signs of acceleration, you know that you're not really plummeting and you refuse to give up Obama's twitter account!

would only apply if your capsule was moving with a constant speed (e.g. being lowered by a giant winch). If Dr. Tide is willing to go to all the trouble to lower you with a perfectly constant velocity into the center of the earth so that you can't tell what's happening, then he probably deserves to beat you. You should just call his bluff—tell him that your superior knowledge of physics tells you that since you don't feel the tell-tale signs of acceleration, you know that you're not really plummeting and you refuse to give up Obama's twitter account!

Q11

So, your friend does not agree with you. She says that the flash farthest from you actually happens first.

If the time difference in your reference frame were large enough then the time difference in your friend's reference frame would become positive and the two observers would agree on the order of the flashes. For example, for 2 s, we get the following:

The key is that whenever distance separating two events satisfies d > ct∗ c two observers v

may disagree on the order of the events, depending on their relative motion. On the

other hand, if the events are separated by d < ct ∗ c the observers will always agree on v

the order no matter what their relative speed.

If the time difference in your reference frame were large enough then the time difference in your friend's reference frame would become positive and the two observers would agree on the order of the flashes. For example, for 2 s, we get the following:

The key is that whenever distance separating two events satisfies d > ct∗ c two observers v

may disagree on the order of the events, depending on their relative motion. On the

other hand, if the events are separated by d < ct ∗ c the observers will always agree on v

the order no matter what their relative speed.

work = force over a distance

force must be directed along the path of motion otherwise no work is done (eg. centripetal spinning)

Based on the prefaces to their books, what points do you think David Griffiths and Brian Greene would agree on?

In fact, they would agree on quite a number of points:

1. The big revolutions of modern physics rest of relativity and quantum mechanics.

2. What we observe in the physical world is critical for separating "facts" from unfounded imagination.

3. Their books are written for the general reader. By implication, this suggests that they also agree that it's important for the general reader to learn about the breakthroughs in modern physics.

1. The big revolutions of modern physics rest of relativity and quantum mechanics.

2. What we observe in the physical world is critical for separating "facts" from unfounded imagination.

3. Their books are written for the general reader. By implication, this suggests that they also agree that it's important for the general reader to learn about the breakthroughs in modern physics.

Based on the prefaces, on what point or points would Brian Greene and David Griffiths disagree?

Again, there one could argue for a number of different answers. Certainly, they disagree on the style of presentation. In fact, Griffiths draws attention to the fact that his book does not follow one of these "NOVA" types of presentation. It also seems that they disagree on what types of material to include, as Griffiths specifically points to string theory as something he won't discuss while Greene devotes a significant part of his book to it. I think you might also argue that they disagree about the role of imagination in science. They certainly both argue that it's important, but Griffiths seems to imply that imagination is only a fall-back when you can't make progress based solely on observation, while I think Greene would say that any theory you can dream up that fits with existing observations and makes predictions about new ones is a good theory.

What are the "two clouds" Lord Kelvin referred to and what revolutions were sparked in answer the issues they raised?

The "two clouds" refer to the problems of understanding (a) the properties of light's motion and (b) radiation emitted by heated objects. The first "cloud" sparked the revolution of relativity, while the second launched quantum mechanics.

Summarize briefly how relativity and quantum mechanics impact our views on space and time.

The insight from relativity is that space and time aren't separate things, but are one connected whole (spacetime). Furthermore, they are flexible and changing, rather than static and rigid.

Quantum mechanics challenges our view that objects separated in space are independent and can't influence one another. In quantum mechanics two objects can be separated by a great distance and yet still be considered one connected object.

Quantum mechanics challenges our view that objects separated in space are independent and can't influence one another. In quantum mechanics two objects can be separated by a great distance and yet still be considered one connected object.

What problem are people trying to solve in searching for a "unified theory"?

They are trying to solve the problems that arise when you try to use quantum mechanics and gravity together. Currently, when used together, these two theories produce nonsense results.

Are speed and velocity different? Explain. How does this relate to acceleration?

Speed and velocity are different. Speed is the magnitude of velocity. In other words, speed tells you how fast something is going while velocity tells you how fast and in what direction. Acceleration is a change in velocity, so an object that has a constant speed can still be accelerating if it is changing direction.

What error did Aristotle make that took approximately 1500 years to correct? Why was it so hard to figure this out?

The error was in believing that for something to move at a constant speed, a force must be supplied. It took 1500 years to sort this out because you don't observe the correct behavior--constant speed coming with no force applied--unless you can eliminate the ubiquitous forces of of friction, air resistance, etc.

Newton's 3rd Law says that for every force applied, there is an equal (in magnitude) and opposite (in direction) force. Why doesn't this mean that nothing can accelerate because all forces are canceled out?

Every force gets paired with an equal magnitude force in the opposite direction, which of course would cancel out mathematically. However, these forces never apply to the same object. In other words if I want to run forward, I exert a force on the ground, and the ground exerts a force on me. These forces would cancel out if they were on the same object and lead to no acceleration, but because they are on different object (me vs the ground) they don't cancel each other, and I can in fact run.

How are Newton's Law of Universal Gravitation and Coulomb's Law similar? How are they different?

Aside from the obvious similarities (they both tell us about forces, etc.), they both take the form where the force is determined by a product of two quantities (one related to each object, like their masses or charges) divided by the square of the distance between them. This similarity is not coincidental, but we won't have time to explore that deeply. There are also many obvious differences (one is for gravity while the other is for electricity), but the most significant difference is that electric charge comes in two types (so you have both attractive and repulsive forces) while mass is always positive (as far as we know) so gravity only attracts and never repels (at least according to Newton...)

Explain the difference between kinetic energy and momentum.

Kinetic energy and momentum both indicate something about motion. They both depend on the speed and mass of the moving object. Kinetic energy goes as the square of the speed, while momentum is linear with speed. More significantly, momentum is related to "how hard it is to stop something." In other words, it's connected with the force you need to exert on an object. On the other hand, kinetic energy represents "the capacity to do work by virtue of motion." This is connected with how much work an object can do (or can have done to it) before coming to rest.

Why does Griffith's give us two equations for gravitational potential energy (V = mgh and V = -GMm/R). Why the negative sign in the second equation?

The first version (V = mgh) is only applicable when the objects remain close to the surface of the earth and we don't need to account for the changing strength of the gravitation force. This effect is accounted for in the second equation. The negative sign is critical because potential energy should increase with height. In the second from, the height effects R, which is in the denominator. Without the negative sign, that equation would have gravitation potential energy decreasing with additional height. Another way to put it is that the negative sign comes from the fact that M and m are attracted to one another, so you have to do work to separate them.

Explain the difference between constructive and destructive interference.

When two different waves overlap in the same region, they "interfere" which basically means you add them together. In the case of constructive interference, the waves are "in phase" meaning the peaks line up and the waves add together. In the case of destructive interference, the waves are out of phase, meaning the peak of one lines up with the trough of the other, and the waves cancel out.

What condition or conditions are required to create a standing wave?

A standing wave occurs when a wave is created in something with endpoints that the wave can reflect from (like the string of a violin or the tube of a clarinet. If the wave has just the right frequency so that the reflections from the end are in phase with each other, so that the reflections constructively interfere, a standing wave is created. In terms of wave lengths, this means the length of the region containing the standing wave must be equal to a multiple of 1/2 the wave's wavelength.

How does Einstein view the connection between geometry and reality? When he defines coordinates, how does he make the connection more direct?

Einstein is careful to point out the the results of geometry, although following logically from the assumed axioms, don't necessarily have anything to do with reality. In order to increase the strength of the connections between geometry and reality, Einstein defines coordinate systems in terms of physical object (an array of standard length rods).

How does Einstein define position and time?

For Einstein, the abstract concept of space is not helpful. Instead, he defines position based on the comparison of an object's location with a hypothetical array of rigid objects of standard length, forming a physical manifestation of a coordinate system. That such an array doesn't actually exist does not eliminate the validity of the idea. Time is also defined based on the motion of clocks held by the observers. The key point here is that for Einstein, position and time are linked to physical objects and have no meaning in their absence.

What is a Galilean coordinate system?

It is a frame of reference in which "the laws of Galilei-Newton hold." In other words, it's a frame of reference in which Newton's three laws of motion are valid. Generally speaking, this means it has to be a frame where a = 0, also referred to as an inertial frame. In frames of reference that accelerate, Newton's laws don't hold: objects accelerate with no obvious forces, they don't necessarily stay in constant motion if not acted on by an external force, etc. This is sometimes referred to as an inertial coordinate system (e.g. in Griffith's book).

What is "the principle of relativity (in the restricted sense)"?

The principle of relativity means that the general laws of physics the apply in one reference frame also apply in any other. It is restricted because this principle (for the moment) only applies to Galilean references frames (that is frames moving at a constant velocity with respect to other frames.

What problem does light pose when considering observers moving with different speeds?

Electromagnetism shows that light propagates at a constant speed of <i>c</i>, making no reference to a specific observer or reference frame. This can be interpreted as all observers measuring the speed of light as <i>c</i>, regardless of their relative motion to other observers. This contradicts our expectations from (Galilean) relativity, which says that if one observer measures the speed of light as <i>c</i> and another observer moving with a speed of <i>v</i> relative to the first, will observe the light moving with a speed of <i>c</i> + <i>v</i>.

What difficulty must be addressed to define the meaning of simultaneous in a rigorous way? How does Einstein define two events that are simultaneous?

The difficulty is the finite speed of light. Signals don't travel from one place to another with infinite speed, so you can't send a message saying "Now!" that will arrive instantly. Einstein's definition of simultaneous accounts for this by saying that two events are simultaneous if a light-speed signal (i.e. a flash of light) arrives at a place physically located half-way between the two events at the same time.

Do two observers always agree on whether events are simultaneous? Do they always agree on the distance separating two objects? When do they disagree?

Observers don't necessarily agree on whether events are simultaneous; nor do they agree on distances measured. Disagreements occur when the observers are in relative motion with respect to each other.

Lorentz Transformations tell how different observers will measure space and time so that they always agree on what quantity?

C

What specific argument does Einstein give for saying that the speed of light must be the fastest speed something can attain?

He points out that in the Lorentz transformation/length contraction, and time dilation equations, using a value for v bigger than 1 results in a square root of a negative number in the equations, suggesting that it's not possible to go faster than the speed of light. He also points out that the energy of a particle approaches infinity as the particle approaches the speed of light, which also prevents particles from exceeding the speed of light.

What role did the Fizeau experiment play in developing the theory of special relativity?

Fizeau's experiment determined that the Lorentz equation for the motion of a point with reference to a system K works in practice much better than the Galilean equation. With regards to the propagation of light, Fizeau's results correspond almost perfectly to the results predicted by relativity.

Model Short Answer: It confirms the addition of velocity equation from special relativity.

Comment: Not just in practice--special relativity's equations are right while Galilean relativity's equations are wrong.

Model Short Answer: It confirms the addition of velocity equation from special relativity.

Comment: Not just in practice--special relativity's equations are right while Galilean relativity's equations are wrong.

Einstein writes that because of relativity, two separate conservation laws: conservation of mass and conservation of energy, are actually the same. What does he mean by this?

Einstein demonstrates that there is energy associated with an objects mass. In relativity, the formula used to calculate kinetic energy does not vanish as v = 0. Rather, it goes to a minimum value of E = mc^2, which is considered the rest (i.e. v = 0) mass of an object.

What is Einstein referring to when he mentions "aberration"?

He is referring to the fact that certain stars appear to change position slightly over the course of the year. This could not be explained until relativity demonstrated it was an effect of the relative motion of the stars compared to the earth.

What is the "logical circle" that Griffith's mentions when defining the principle of relativity and how does he break it?

The logical circle is that in defining the principle of relativity, we refer to a system that's "at rest," but it's impossible to say any system is at rest, since all motion is relative. He breaks out of the circle by changing the definition of the principle of relativity to instead say that the laws of physics are the same for any inertial (i.e. Galilean) coordinate system.

Why did no one before Einstein notice that the Galilean Velocity Addition rule is not correct but needs to be modified as Einstein did?

This is because in Einstein's rule, the regular Galilean formula is divided by (1 + v_AB * v_BC/c^2). Under most ordinary circumstance, both v_AB and v_BC are much smaller than c, so this factor is very close to 1 and Einstein's and Galileo's equations give nearly the same answer. It's only when speeds approach c that noticeable differences occur.

Explain what Griffiths means when he says, "What you observe, then, is not at all the same as what you see and hear."

He means that with seeing and hearing there are intrinsic delays that deal with the propagation time of light and sound that have nothing to do with relativistic effects. Observations have been corrected for all such propagation delays.

Is it correct to say that moving objects appear shorter and moving clocks appear to run slow? Explain.

No. Relativity is dealing with what we observe (get after accounting for time), not what appears to be happening. When moving, objects are actually shorter and time does actually run slower. This is so much more definitive than what appears to be and that's a big reason why it's so astounding.

Model Short Answer: This sounds right, but the word "appear" makes it tricky. People usually say "appear" to indicate that something seems to be one way, but in fact, it's not. Think "appearances can be deceiving." In relativity, there is no deception. Moving objects really are shorter and moving clocks really do run slower than when at rest.

Model Short Answer: This sounds right, but the word "appear" makes it tricky. People usually say "appear" to indicate that something seems to be one way, but in fact, it's not. Think "appearances can be deceiving." In relativity, there is no deception. Moving objects really are shorter and moving clocks really do run slower than when at rest.

What do the paradox of time dilation and the paradox of Lorentz contraction have in common?

They both involve what appears to be a contradiction between two observers viewpoints. Intuitively, we think that if person A's clocks run slower or rods look shorter according to person B, the person B's clocks would seem to run faster and rods would look longer to person A. However, on closer reflection, this wouldn't make sense. The laws of physics have to be the same for everyone, regardless of the state of motion: So if moving clocks runs slower and moving rods look shorter to one person, they have to run slower or look shorter to both.

What surprising thing happens during the collision between the two lumps of clay in Example 4?

After the collision, the combined mass of the two lumps of clay is more than the sum of the individual lumps before the collision. This is because some of the kinetic energy of the incoming lump has been converted into mass (i.e. rest energy).

What is the loophole that applies to energy and momentum for massless particles?

The formulas for energy and momentum are proportional to mass, so the energy and momentum should go to zero for massless particles. However, if any particle is moving at the speed of light, it's energy and momentum will go to infinity. Due to some mathematical technicalities, zero times infinity can be anything, including a finite number. So it turns out that massless particles must go at the speed of light in order to have a non-zero energy.

What is the difference between the past, the future, and "elsewhere"?

The past includes all points in spacetime that could possibly have had an influence on you now. The future is defined as all points in spacetime that you could possibly have any influence on. That leaves elsewhere, which are all points in spacetime that can't have influenced you now and won't influence your future.

What distinguishes the general theory of relativity from the special one?

The general theory of relativity is valid for all moving coordinate systems, not just ones moving with constant velocity. As such, it needs to incorporate not only acceleration but also gravity.

Explain how a person in a windowless room would distinguish between uniform acceleration in one direction or a uniform gravitational field in the other?

The observer in fact cannot distinguish. A uniform gravitational field pointing to the left is indistinguishable from constant acceleration to the right. This is the foundation for general relativity.

What happens to the path of a beam of light traveling past an object with a gravitational field?

The gravitational field will cause the light ray to bend in its path.

What happens to a system of measuring rods and clocks if put into rotation?

Rods placed along the direction of motion will be shortened compared to rods oriented perpendicular to the direction. Thus rods used to measure the circumference of a circle the rods will be shortened, requiring more of them to cover the circumference of the circle than would be needed in the absence of rotation. However, along the radial direction there would be no shortening (since the radial direction is perpendicular to the motion), so (circumference)/(2*radius) > pi. Likewise clocks placed further from the center of rotation would run slower than clocks placed near the center.

-light acts like a particle with a frequency

2 big things of relativity:

-speed of light is constant for everyone

-laws of physics are different for everyone

2 big things of relativity:

-speed of light is constant for everyone

-laws of physics are different for everyone

...

e=mc^2/sqrt(1-v^2/c^2)

denominator adds extra energy from motion

denominator adds extra energy from motion

...

Lorentz contraction paradox:

lengths of a moving object appear shorter as do lengths of the rest system with respect to the mover

time dilation paradox:

time is shorter for observer's measurement of mover and vice versa

barn paradox:

guy running w/ ladder: barn shrinks, ladder same, won't fit

guy at rest, closing door: ladder shrinks, barn same, it fits

how? rest guy sees door close, moving sees ladder first b/c of light. they just disagree on the order of events

guy at rest, closing door: ladder shrinks, barn same, it fits

how? rest guy sees door close, moving sees ladder first b/c of light. they just disagree on the order of events

twin paradox:

time is moving more slowly on the ship according to the twin on Earth. the ship reaches the destination, and slows down for a time to turn around. on return, clock runs slow again. according to twin on earth, the traveller ages more. but from twin's perspective on ship, time on earth is passing slower also! how are these reconcilable? when the mover gets back, he expects to be older. but the person on the rocket accelerated at the destination, and special relativity doesn't hold anymore. so the mover has aged less since there's no competing explanation.

Exam2

What did Planck, Einstein, and Compton teach us about light?

What did Planck, Einstein, and Compton teach us about light?

All three contributed to our understanding that light behaves like a particle (i.e. photon). Planck is the first one who was able to solve a problem (blackbody radiation) by treating light as if it comes in little bits of energy. Einstein took the idea to its full conclusion and used it to explain the photoelectric effect. (Planck thought light might only behave like a particle for certain special cases.) Compton showed that you could only understand certain phenomena involving light scattering off individual electrons if you treat the light as a particle and use the standard conservation of energy and momentum equations to get the answer.

What new idea did de Broglie introduce?

De Broglie, building on the work of Planck, Einstein, Compton, and others, introduced the idea that particles behave like waves with the wavelength given by h/p.

Describe the key elements in Bohr's model of the hydrogen atom.

Bohr pictured the hydrogen atom as an electron orbiting a proton, like a moon orbiting a planet. Furthermore, he inferred that the electron orbits had to be spaced such that exactly a whole number of photon wavelengths would fit around the orbit. This limits the orbital radii (and therefore the energy) of the electron in the atom. Interestingly enough, Bohr cobbled this model together, playing fast-and-loose with the laws of physics and without a full understanding of quantum mechanics, and yet it basically gives the same answers as the full theory.

Why do the dark lines in the absorption spectrum and the bright lines in the emission spectrum only occur for certain wavelengths? Can you explain this in terms of Bohr's model of hydrogen?

The dark lines in the absorption spectrum correspond to wavelengths (and therefore also frequencies) of light that are absorbed when hydrogen is illuminated with light of all frequencies. Likewise, the bright lines in the emissions spectrum show that only light of particular wavelengths (and again therefore also frequencies) is emitted when hydrogen gas is heated. Taken together, these two spectra demonstrate that hydrogen atoms will only absorb or emit light with very particular wavelengths. This make sense because for light to be absorbed or emitted, the electron in the hydrogen atom has to change its energy. In Bohr's model, the electron can only have certain specific energies (often called energy levels), and so the only light that can be absorbed or emitted has to correspond in energy exactly to the difference between two levels. Note: Only one photon can be absorbed or emitted at the same time. Because the energy of a photon is connected with its frequency (and therefor wavelength), this means that only certain wavelength can be absorbed or emitted.

What does a wave function tell us about a particle?

The wave function (or more specifically the square of the wave function) tells us about the probability of finding a particle in a given location.

How is the wave nature of particles connected with Heisenberg's uncertainty principle?

Momentum is tied to the wavelength of a wave. To have the most certain determination of a wave's wavelength, you need a wave that spreads out throughout all of space. Then you won't be able to determine the position of the particle associated with that wave at all. On the other hand, to have a wave with a definite position, it needs to be very localized in space. For such a wave, it is very difficult to determine the wavelength with any certainty. So, the complementary nature of position and wavelength certainty for a wave translates directly into momentum versus position uncertainty in Heisenberg's uncertainty principle.

Which of the three schools of thought on quantum mechanics seems most likely to you? Explain why you feel that way.

Aside from the evidence pointing towards the orthodox interpretation, it seems like the most likely explanation to me because of the uncertainty principle. Since the act of measuring a particle has an effect on it (like the example of hitting it with a photon) then it seems to make sense that measuring it forces it to take a stand and become clearly defined. Also, since the double-slit experiment shows that measuring a particle induces the collapse of the interference pattern, the orthodox response seems to best explain that.

What is an entangled state?

In quantum mechanics, an entangled state is a special arrangement of two particles where before you make a measurement, you don't know the state of either particle, but after you measure one particle, you know precisely the state of both particles.

What conundrum is posed by Newton's bucket?

Because the water in the spinning bucket eventually takes on a curved shape, the questions is, how does the water know that it's spinning? Is it because it's spinning relative to something physical, or is it because, for example, space is an absolute with respect to which the bucket can spin.

Compare Newton's, Leibniz's, and Mach's view on space?

Newton sees space as having an absolute, independent existence. Leibniz instead sees space as being completely relative to things in space. Mach takes a view similar to Leibniz, going so far as to say the sorts of forces we experience from things like spinning or acceleration only happen if there is other matter in the universe relative to which we are accelerating.

For Einstein, does space or time have an absolute existence?

Einstein does not view space or time as absolute. But spacetime does end up being absolute. Acceleration can be judged relative to absolute spacetime depending on whether the path of an object has a straight line in spacetime or not.

Explain the "loaf" analogy.

Greene describes how different observers view "now" in space time by likening it to slicing a loaf of bread. Each slice determines what events in space are considered to be happening at the same time. However, different observers slice the loaf at different angles (or in general relativity, even in warped slices), and so don't agree on which things are happening at the same time. However, all observers will agree that when the put all the slices together, they get the same loaf.

According to Einstein's general relativity, which person would be most accurately described as not accelerating: Someone sitting in a chair on the surface of the earth or someone falling freely towards the surface of the earth from a great height. Ignore the motion (spinning, orbiting, etc. of the earth, as well as the effects of air resistance.)

The person falling freely towards earth is actually the one not accelerating. The person in the chair (and every object on earth) is accelerating upward all the time. They feel the effects of gravity and thus cannot be the benchmark against which to judge all other motion. The person in true free-fall feels no effects of gravity - it would feel the same as freely floating in space. Since they feel no effects of gravity, they are actually not accelerating, and are the benchmark to judge motion against.

What is the difference between probability, as it is used for predicting weather or gambling, and probability as it is used in quantum mechanics?

In meteorology or gambling, probability is used in predictions because of a lack of information. You can't say precisely whether it will rain, or what numbers will show on the dice you roll because you don't know everything about every particle in the atmosphere or about the dice as they leave your hand and land on the table. If we did know everything though, we could use Newton's laws, etc. to predict the exact outcome. In quantum mechanics, even if you know everything about a particular experiment, the results can only be predicted in terms of probability. Probability is fundamental to the description--as encoded in the wave function--and no amount of additional knowledge will turn quantum mechanic's wave functions into definite predictions.

What feature of the quantum mechanics was the argument proposed by Einstein, Podalski, and Rosen (sometimes also called the EPR paradox) trying to attack? How did the basic reasoning of the argument go?

EPR were trying to refute the Heisenberg uncertainty principle that states that any particular quantum object cannot simultaneously have a precisely determined position and momentum. The basic gist of the argument was that using a pair of entangled particles, you could make a momentum measurement on one (thereby determining the momentum of the other) and then measure the position of the other particle. Combining the two measurements would give you an arbitrarily precise measurement of both position and momentum that could evade the uncertainty principle. Key to this argument was the assumption that if the two entangled particles are far apart, a measurement on one can't affect the other one.

How is the argument between EPR and the defenders of quantum mechanics (Bohr, Pauli, etc.) similar to debate between Newton and Leibniz about the nature of empty space? What's the bucket for EPR vs quantum mechanics?

The similarity is that in both cases, people involved in the argument believed strongly in their perspective, without having any way to test which viewpoint was right. In Newton vs Leibniz, the critical turning point was when Newton introduced the idea of the spinning bucket as a way to see that "empty space" had observable consequences. For EPR vs QM, the "bucket" was Bell's Theorem which suggests a way to test the difference in EPR's and QM's perspectives.

What is the spin of a particle? In quantum mechanics, how is the spin of a particle about one spatial axis related to the spin around the other axes?

All particles behave as if they are spinning around some axis (i.e. direction). This spinning about an arbitrary direction can be described generally as some amount of spinning around each of the three spatial axes (e.g. x, y, and z). However, in QM, if you try to measure the amount of the spin around some axis, you will find that all of the spinning is around the axis you try to measure, and you will lose information about how much spinning may have been around the other axes.

What assumption that is part of the EPR argument was shown to be wrong?

The refuted assumption is that of locality--that measurements on a particle in one place wouldn't affect the entangled particle somewhere else.

Why doesn't the connection between two entangled particles cause problems with relativity (at least in the "majority" view--ignoring the "contrarian" view)?

The main concern is that "something" has to pass instantaneously between the two entangled particles when a measurement is made. However, that something does not correspond to energy, momentum, or even information. Most people believe that the speed limit in special relativity only corresponds to things that could be used to transmit a message (i.e. information), and so the instantaneous effect of making a measurement on one of the two particles doesn't constitute a violation of special relativity.

How does physics (in particular relativity) distinguish between past, "now", and future?

In some sense, the concept of "now" does not exist in relativity. Time behaves just like the spatial dimensions. It's like an axis, and any point along the axis is equivalent to any other point.

What is a "now list"? Do different observers agree on their now lists? What factors might influence this?

A "now list" is a mental image of all the things that exist in your "slice" of the space time loaf. Different observers will slice the loaf different ways, so their now lists will not agree with yours. In other words, they will list as "now" things that you might say are in the past or future. There are really two effects that influence how different your now list is from another's: Your relative speed and the distance between you and the other observer. The speed effects the angle between your slice and another observers, while the distance magnifies the size of the effect.

What is "time reversal" symmetry? Please give an example from the reading. Is time reversal a good symmetry for the laws of physics?

Time reversal symmetry refers to the idea that for any particular motion, if you ran that motion "backwards in time," like running a film backwards, the reversed motion would also be allowed by the laws of physics. The example given in the book is a tennis ball hit from Venus to Jupiter. Reversing the motion (from Jupiter to Venus) even accounting for the affects of Venus' and Jupiter's gravity, etc. is allowed by physics. Time reversal symmetry is a part of all physical laws (except maybe Quantum Mechanics, which we won't address until next chapter.

What is entropy? What is the distinction between having high entropy and low entropy?

Entropy refers to the number of possible equivalent configurations for a system. Having high entropy means that there are many ways to arrange your system (pages in a book, atoms in a gas, etc.) that would go unnoticed. Low entropy means that there are not many unnoticeable rearrangements. High entropy is generally associated with being disordered because it is more difficult to spot rearrangements in a disordered system.

What is the Second Law of Thermodynamics? When combined with time reversal symmetry in physics, what is the surprising conclusion?

The Second Law of Thermodynamics says that systems tend to evolve from lower entropy to higher entropy. However, since the laws of physics are symmetric under time reversal (that is to say that they don't prefer on direction of evolution in time to another), the surprising conclusion is that entropy should increase not only when evolving a system forward in time, but also when evolving it backwards in time!

Is the formation of highly ordered stars from the diffuse gas of the early universe a violation of the Second Law of Thermodynamics? Explain.

Although stars have low entropy, the net entropy of the universe after star formation is not lower than the entropy of the early universe consisting only of gas. This is because the entropy lost when stars form is more than compensated for by the entropy gained as the stars give off energy in the form of light and heat, causing the entropy of the rest of the universe to increase enough that the net entropy of the whole universe increases.

How does the quantum concept of "the past" or history differ from the classical one?

In classical physics, the past is unique--an object arrives at location A by traveling a unique and definite path from it's origin to that location. In contrast, in quantum mechanics, the past is not unique. A given observation has a probability that can only be calculated by considering all possible pasts that could have generated the observed outcome. For example, in the case of two slit interference, you have to treat each photon as if it went through both slits, not just one or the other as would be the case for classical physics.

Why do objects on "everyday" scales behave as though they have a unique history while microscopic objects do not?

For macroscopic objects, the process of "averaging" over possible paths is dominated by the path that classical physics would say is "the path." In other words, one path contributes far more than the other paths. But for small objects, like an electron, there are a larger number of possible paths/histories that give a contribution, making the behavior less classical and more quantum.

What is John Wheeler's Delayed Choice experiment and how does it challenge our classical conception of past vs present?

Wheeler's Delayed Choice experiment is a variation on the "two slit" or "two path" experiments where a detector is added to one path. The detector, if turned on, will register which path the particle has traveled, thereby eliminating the interference pattern. The wrinkle is that the detector is located so that when the detector is turned on, the particles that will be encountering it had to make a decision a long time ago (at the location of the slits or the beam splitter) as to whether they would choose to travel a definite path or a superposition of paths. This makes it seem as though an act in the present (turning on the detector) changes the outcome of something in the past (determining which path to take.

In the delayed-choice quantum eraser experiment, what determines whether a given photon shows evidence of interference?

If the idler photon corresponding to the given photon traveled along a path causing it to be registered in a detector the definitively determines which path the original photon took to the screen, there will be no interference. But if the idler photon travels along a path that makes it impossible to tell the original photon's path, the given photon will interfere. For such photons, when a large number of them have been recorded, an interference pattern will be visible.

If there is an "arrow" of time in quantum mechanics, what aspect of quantum mechanics must be involved?

The arrow of time would have to come in through the process of measurement, because the Schödinger equation definitely does not differentiate between backwards and forwards in time.

What are the four interpretations of the quantum measurement problem does Greene discuss? Please give a sentence describing each.

Heisenberg: Wave functions represent knowledge, and the collapse of the wave function represents only a change in what we know.

Everett, Many Worlds: Wave functions never collapse. Every possibility represented in the wave function comes to exist in its own parallel universe.

Bohm: The wavefunction is a separate entity from the particles it's associated with. Particles take definite trajectories, but wavefunctions behave like quantum objects. Effects that influence a wavefunction in one place can instantaneously propagate to particles far away. This theory is a non-local hidden variable theory.

Ghirardi, Rimini, and Weber: This interpretation modifies Schrödinger's equation to include an explicit mechanism for the spontaneous collapse of a wavefunction. For a single particle this rarely happens, but for large collections, one or more particles is likely to collapse, trigger a chain reaction causing the wavefunctions of all the other particles also to collapse.

Everett, Many Worlds: Wave functions never collapse. Every possibility represented in the wave function comes to exist in its own parallel universe.

Bohm: The wavefunction is a separate entity from the particles it's associated with. Particles take definite trajectories, but wavefunctions behave like quantum objects. Effects that influence a wavefunction in one place can instantaneously propagate to particles far away. This theory is a non-local hidden variable theory.

Ghirardi, Rimini, and Weber: This interpretation modifies Schrödinger's equation to include an explicit mechanism for the spontaneous collapse of a wavefunction. For a single particle this rarely happens, but for large collections, one or more particles is likely to collapse, trigger a chain reaction causing the wavefunctions of all the other particles also to collapse.

How does decoherence address the quantum measurement problem?

Decoherence, the random interaction of the environment with the quantum system that spoils the coherence necessary to observe quantum effects like interference, resolves the measurement problem by removing "spooky" quantum effects like interference and making quantum probabilities behave just like classical ones.

Why don't all the electrons in an atom like carbon (which has 6 electrons) occupy the lowest energy level (n = 1)?

The Pauli Exclusion Principle says that two electrons cannot occupy the same state. It turns out that at there are two "states" corresponding to the n=1 ground state. (The two states in fact correspond to the electrons spinning "clockwise" and "counterclockwise," though Griffiths doesn't explain that. So, only two electrons fit in the n = 1 ground state, and others fill in higher states.

What determines how chemically reactive an atom is?

How chemically reactive an atom is is determined by the number of electrons it has in its outermost, not completely filled energy levels of shells. These electrons are called valence electrons. In atoms where the outermost shell is completely filled, the atom is not very reactive at all.

What are isotopes?

Isotopes are atoms with the same number of protons but different numbers of neutrons. Because the number of protons determines what element an atom is, isotopes all belong to the same element and have the same chemical properties, but with different numbers neutrons they have different masses.

What is the difference between fission and fusion? Why do very heavy nuclei release energy during fission, but for light nuclei, fusion is what generates energy?

Fission is the breaking apart of a nucleus into two lighter nuclei while fusion is the merging together of two nuclei to form one heavier one. Whether fission or fusion releases energy or absorbs it depends on whether the total mass of the products after fission or fusion weight more or less than the nuclei before the process, which is determined by the binding energy of the nucleus. The element with the highest binding energy per nucleon is Iron (Fe). Element heavier than iron will release energy when they fission because they will be forming nuclei that are closer to iron. For elements lighter than iron, they will release energy during fusion. For heavy elements, it takes a net input of energy to make fusion happen, and for lighter element, energy must be supplied (instead of being generated) to cause fission to occur.

How did Pauli deduce that a new particle (which was later named the neutrino) must exist?

He deduced this by studying beta decay. Originally, scientists thought the beta decay process was n decays to p + e. However, if this were true, energy and momentum conservation demand that every electron originating from beta decay should have the same energy and momentum. What is observed instead is that the electrons have a range of energy and momenta. Pauli reasoned that this situation could arise if there was an extra particle, with low mass so that it wouldn't screw up our expectations about the maximum possible electron energy, and electrically neutral so that it couldn't be detected by electromagnetic interactions. Thus, the beta decay process becomes n -> p + e + nu, and the energy and momentum are shared among the three decay products.

What led Yukawa to hypothesize that mesons should exist?

He was studying the strong force. He reasoned that like the electromagnetic force has the photon, the strong force should have a mediator particle. Based on the short range of the force, he knew the carrier of the strong force would have to be massive. He was able to calculate that the mass should be between an electron and a proton--being in the middle, he called it a meson.

In the quark model, what is the key distinction between the particles we call baryons and those we call mesons?

Quark content is the key distinction. Baryons consist of either three quarks or three antiquarks. In contrast, mesons are always composed of exactly one quark plus one antiquark. These are the only two configurations that quarks assemble in because they are the only two ways to generate a "white" or colorless object from color quarks.

Why don't we observe free quarks in nature?

Free quarks are not seen in nature because of quark confinement. This idea states that only colorless (or "white") particles can exist freely in nature. Because quarks carry color, they cannot exist alone, but must group together in combination that have no net color (either one of each color, or color + anti-color).

What discovery launched the November Revolution?

The discovery of the psi meson (usually referred to in other texts as the J/psi). This meson contains a charm quark and an charm antiquark. Although the existence of the charm quark had been already predicted by theorists, it had not yet been observed. The discovery of the J/psi became the turning point in the development of what we now call the Standard Model.

Please list the four fundamental forces and for each state the carrier particle and which particles from the Standard Model experience these forces.

Strong - gluons(8) - baryons and mesons experience Electromagnetic - photon - holds atoms together

Weak - W and Z bosons - neutrino interactions

Gravitational - graviton - large scale structure

Weak - W and Z bosons - neutrino interactions

Gravitational - graviton - large scale structure

How is an antiparticle different from its corresponding particle?

An antiparticle is different from its corresponding particle in just about every way that matters: It has electric and also color charge. If the particles counts for "+1" of a specific type (e.g. strangeness, baryon, lepton, etc.) the antiparticle will count as "-1." The only way in which the antiparticle is not the opposite of the particle is in terms of the mass. Particles and antiparticles have the same mass.

Which of the conservations laws listed by Griffiths is only "approximate." Explain.

The quark flavor conservation law is only approximate because charged weak interactions change quark flavor. However, all other interactions preserve quark flavor, and since the weak interaction is so much smaller than the others, it is often a good approximation to treat quark flavor as conserved.

What is a cosmological redshift? What does it mean that the light from all distant objects around us is redshifted?

A cosmological redshift comes from light becoming stretched out when emitted from receding objects. The fact that all the light we see from distant objects is redshifted tells us that everything is moving away from us, because of the expansion of the universe.

Where does cosmic microwave background radiation come from? What did Penzias and Wilson originally think was causing them to detect this radiation?

The cosmic microwave background comes from photons liberated in the early universe when the temperature cooled enough the ions zipping around formed neutral atoms and the universe become transparent. However, originally Penzias and Wilson thought they were just seeing the effects of bird droppings on their measurement equipment.

Where do elements heavier than lithium come from?

The heavier elements (including things like carbon, oxygen, etc. that make up us) are created from the nuclear reactions that occur inside a star (including the process of the star going supernova).

How do we "see" dark matter?

We see dark matter through its gravitational affects on other things. Dark matter impacts the way galaxies rotate and even alters the path of light causing images of stars and galaxies to be warped in ways that let us measure the amount of dark matter present.

What determines the shape of the universe?

The shape of the universe is determined by the density of matter. If the density is too high, the universe "closes" on itself and has a spherical shape, while too low a density produces as hyperbolic universe. Exactly the right critical density gives a flat universe.

In terms of space, define what a symmetry is? How is the concept of symmetry applied to physics in general?

A spatial symmetry is a set of spatial manipulations (rotations, translations, etc.) that leave the appearance of an object unchanged. More generally, in physics a symmetry is a set of manipulations that leave the laws of physics unchanged (although the observations of a particular observer might be different).

Why are we justified in claiming that the universe is 14 billion years old, without specifying the specific observer that such a time measurement corresponds to?

We can talk about an absolute age of the universe because space is spatially symmetric and uniform. Because of this uniformity, clocks in different parts of the universe experience the same gravitational fields, etc. and therefore run at more or less the same rate. We know that the universe is spatially uniform, in part, because of cosmic microwave background radiation, which is extremely uniform in temperature.

How does the expansion of the universe differ from a conventional explosion?

In a conventional explosion, objects are moving away from each other through space, and by tracing their paths through space backwards, you will find the origin of the explosion. The expansion of the universe on the other hand is an expansion of space itself. As a result, this expansion is completely symmetric in space--there is no special center or origin for this expansion.

What are the three possible shapes the universe can take? Please give examples of each shape (the 2-D versions are fine). Which (if any) of these shapes can have a finite size?

The three shapes can be defined by their curvature: positive, zero, or negative. A shape with positive curvature is like a sphere in 2D. A shape with zero curvature is flat, like a plane. A shape with negative curvature is like a saddle shape (or a Pringles Chip) in 2D. All three of these shapes can be of finite size. The positive shape is always of finite size. The flat or saddle (negative or zero curvature) shapes can be made finite size by letting their opposite edges wrap around and connect, like the edges of a video game screen.

How does our picture of the big bang change if we treat the size of space as infinite instead of finite?

If space is infinite in size, then we can't think of it as being larger or smaller at later or earlier times. The expansion of space makes everything in space get farther apart, but space doesn't get any larger, since two times infinity (for example) is still infinity. In this picture, the big bang didn't start as small point, but instead was an large density everywhere in an infinitely large space.

What is a phase transition and what does it have to do with symmetry?

A phase transition occurs when adding or subtracting heat energy to a system. At specific temperatures, the structure of the substance undergoes significant changes--for example when ice melts to water or water vaporizes to steam. In phase transitions, the amount of symmetry in the substances changes significantly, increasing or decreasing. For example, ice has fewer symmetries than liquid water.

How is the Higgs field different from other fields, like the electromagnetic field?

Unlike other fields, the Higgs field, at low temperatures, settles to a non-zero value. This is because the potential energy curve of the Higgs field has a funny shape, like a bowl with a bump in the middle, causing a Higgs field value of zero to have more energy than a value greater than zero. This means that in the vacuum (i.e. empty space) there is a Higgs ocean, as if the Higgs field has condensed.

How is the Higgs field connected to mass? How was this situation different in the early universe (before 10-11 ATB)?

The Higgs ocean provides resistance to accelerated motion for fundamental particles that interact with it (basically all the quarks and leptons, plus the W and Z particles). In this way, it acts like a molasses, providing resistance to motion, or like paparazzi that make it hard for particles to move around. (The second analogy is descriptive because it also explains how more popular particles--ones with which the Higgs field interacts more strong--have their motions impeded more and therefore have a higher mass. In the early universe, the temperature was high enough that the Higgs ocean was evaporated, and fundamental particles were all massless, like the photon.

Compare and contrast the Higgs ocean and the Aether?

Like the Aether, the Higgs ocean is an "invisible something" that fills all space, even empty space. Unlike the Aether, the Higgs ocean has no effect on the propagation of light and does not affect constant velocity motion (so the experiments that ruled out the Aether don't preclude the Higgs). There are experimental consequences of the Higgs ocean (like there were for the Aether) and these are being tested at the LHC.

How can the Higgs field be like the cosmological constant? How is it different?

The Higgs field, if it gets caught on the central "bump" of its potential as it cools, will create negative pressure, acting just like a cosmological constant. However, there are key differences. For one, the value won't be constant; it will change as the Higgs field eventually rolls into the valley. In addition, while the cosmological constant was chosen to balance gravity and make a static universe, the equivalent cosmological constant value associated with the Higgs field is~10100 times larger, driving outward expansion instead of a static universe.

What is the horizon problem?

It is the problem that regions of space that are always far enough separated that they wouldn't have had time to interact with one another (because the separation is too great for light to travel in the available time) have the same temperature as measured by the cosmic microwave background radiation.

What is the flatness problem?

The flatness problem involves the fact that to be geometrically flat, the universe must have a particular density called the critical density. If the universe has exactly the critical density at the time of the big bang, it will keep the necessary density and remain flat. However, if it has even a minuscule amount of density more or less, then during the course of the expansion of the universe, the density would change significantly and the universe would no longer be flat today.

What is meant by a standard candle in astronomy and what object ends up serving that purpose?

A standard candle is something with a well understood brightness so its apparent brightness can be used to judge its distance. In Astronomy, type Ia supernova fill this role.

How much of the matter necessary to have the critical density for a flat universe is observable by means of visible light? Where does the rest of the necessary mass/energy come from?

Looking at the stars and accounting for that mass, we find only about 5% of the necessary mass for the critical density. The other 95% comes from dark matter and dark energy. Dark matter is a nonluminous matter that surrounds galaxies. We know it exists because of its gravitational effects on the motion of stars and galaxies. It accounts for 25% of the critical density. The final 70% comes from dark energy. Dark energy is basically a cosmological constant that is causing the expansion of the universe to accelerate. The 70% figure comes from figuring out how much dark energy we need to explain the amount of the expansion seen today.

According to inflationary cosmology, where did the initial inhomogeneities that seeded cosmic structure formation come from?

There came from quantum fluctuations. Because of the uncertainty principle, the fields describing matter and forces in the early universe were under constant fluctuation. When inflation dramatically expanded the size of the universe, these microscopic fluctuations became macroscopic, forming the seeds for the first stars and galaxies.

How does inflationary cosmology make it easier to explain where all the mass and energy needed to create the universe comes from?

Inflationary cosmology has the feature that as space expands, the negative pressure of the inflaton field allows it to "mine" energy from gravity. Therefore, after the rapid expansion of the universe through inflation, the inflaton field's energy has increased enormously (~1090 times), and after inflation, as the inflaton field "rolls" down to its minimum energy state, this energy is converted into matter and radiation. With only 20 lbs worth of inflaton field in a tiny region of space, inflationary cosmology shows that the entire visible universe can be created.

If inflation smoothed out gravitational wrinkles, reducing the amount of entropy in the universe associated with gravity, why isn't this a violation of the 2nd Law of Thermodynamics?

It's not a violation because at the same time it was smoothing out gravitational wrinkles, inflation was creating a huge number of matter and radiation particles. All these particles dramatically increased the entropy in the universe. However, inflation guarantees that the entropy goes up much less than it might otherwise have for the configuration the universe ends up in.

Compare Boltzmann's version of the universe arising as a statistical fluctuation from a high entropy state to a lower one with the inflationary version.

In Boltzmann's version, for our current universe to arise as a statistical fluctuation from a higher entropy state, the most likely scenario is that it arose just moments ago. This is because this would entail the smallest deviation from a high entropy state. If it happened any further back in time than just instants ago, it would require an even larger fluctuation, since, from our memories and reasoning, entropy was lower further back in time. This means all our memories are fabricated as part of this fluctuation, making our experience of reality completely unreliable. In contrast, inflationary cosmology says a tiny fluctuation in entropy in the early universe (the inflaton field getting caught in a small region a space, drives the creation of the entire, higher entropy universe today. This theory is nice because it doesn't destroy our ability to rely on the memories and reasoning that got us to this point in the first place.

What is the main limitation of inflationary cosmology's ability to explain the origin of the universe?

It relies on an assumption about the initial conditions of the universe. It assumes that the preinflationary universe was a chaotic, fluctuating place where the inflaton field could sometimes randomly assume the needed value to drive inflation. We currently lack a theory that can help us understand whether such conditions make sense, although string theory might provide such a theory.

Can we detect the effects of vacuum fluctuations? If so, how?

Yes. In 1948, Dutch physicist Hendrik Casimir noted that two metal plates placed close together would limit the types of electromagnetic fluctuations of the vacuum, creating an imbalance between the fluctuations inside and outside the plates. This imbalance generates a force that can be detected. Various physicists have made measurements to verify Casimir's prediction, including Steve Lamoreaux from University of Washington in 1997 who measured the Casimir force to 5%.

What problem(s) arise when you try to combine general relativity and quantum mechanics?

Quantum fluctuations in space and time that arise at very small length scales and very small time scales destroy all meaning for concepts like left, right, up, down, before, or after. Attempts to calculate any quantity under these chaotic conditions yields infinity.

In "a nutshell" (i.e. concisely) describe what String Theory is.

String theory says that the fundamental, indivisible unit of matter or energy is a tiny vibrating loop of "string." Different vibration patterns produce fundamental particles with different masses and properties.

How does string theory resolve the conflict between general relativity and quantum mechanics?

Because all fundamental particles, including force carriers like gravitons have a non-zero size (set by the size of the loop of string), then it doesn't make sense to consider arbitrarily short distances and times, where quantum mechanics and general relativity cause spacetime to go haywire. By setting a minimum length scale around the plank length, string theory keeps the quantum fluctuations in spacetime at a manageable level.

In order to make quantum mechanics consistent with general relativity, what additional feature about the universe does string theory need to assume? How might this feature remain hidden in everyday experience?

Unification of general relativity and quantum mechanics only works mathematically if there are 9 spatial and one time dimension, not three space and one time. This unexpected requirement requires some work to explain, but it turns out that it's possible that the universe we live in really does have 10 spacetime dimensions. The extra dimensions can be hidden because they might be curled up into such a small size that they can't be noticed for objects of "regular" size (like people).

Give one effect from General Relativity that would support a "Machian" view of space, and one that soundly contradicts it.

Frame dragging falls in line with Mach's view of space because the relative motion (e.g. spinning) of two objects is the most important feature in determining the effects. In fact, based on the ideas of frame dragging, spinning Newton's bucket, or leaving the bucket stationary and spinning the universe would produce identical effects. (However, it should be noted that the explanation for frame dragging--that space is "swirling"--is decidedly non-Machian.) The GR effect that most starkly contradicts Mach however, is that of gravitational waves, since it's actually the fabric of spacetime that is "waving" for these waves.

Give at least three experimental signatures we might hope to see if there are extra dimensions to space that are larger than the Plank scale.

There are a number of possible signatures:

At very small distance scales, gravity's strength would increase faster than 1/r2

Microscopic black holes could be produced either in laboratory or cosmic ray particle collisions, and detected at the LHC or the Pierre Auger observatory

Gravitons produced in collisions at the LHC could "slip through the cracks" left by extra dimensions and carry energy away, leading to the appearance of energy non-conservation in LHC collisions

If the extra dimensions are large enough, they might lead to string excitation energies being low enough that new mass states could be excited in LHC collisions.

At very small distance scales, gravity's strength would increase faster than 1/r2

Microscopic black holes could be produced either in laboratory or cosmic ray particle collisions, and detected at the LHC or the Pierre Auger observatory

Gravitons produced in collisions at the LHC could "slip through the cracks" left by extra dimensions and carry energy away, leading to the appearance of energy non-conservation in LHC collisions

If the extra dimensions are large enough, they might lead to string excitation energies being low enough that new mass states could be excited in LHC collisions.

Aside from providing insights into the homogeneity of the universe or teaching us about quantum fluctuations blown by inflation, list at least two other things we might learn by studying the cosmic microwave background radiation.

The CMB is a rich source of information about the early universe:

The details of the CMB temperature fluctuations can help us decide between the many alternative versions of inflation

Primordial gravitational waves left over from the Big Bang, but to weak to be detected by experiments like LIGO could be imprinted on the CMB polarization.

The shape of the universe (spherical, flat, or hyperbolic) would distort the CMB fluctuations in a noticeable way

The details of the CMB temperature fluctuations can help us decide between the many alternative versions of inflation

Primordial gravitational waves left over from the Big Bang, but to weak to be detected by experiments like LIGO could be imprinted on the CMB polarization.

The shape of the universe (spherical, flat, or hyperbolic) would distort the CMB fluctuations in a noticeable way

What is quintessence?

Quintessence deals with the idea that the current acceleration of the expansion of the universe doesn't come from a constant Dark Energy/cosmological constant, but rather from the inflaton field remaining stuck away from its minimum energy configuration for much longer than in standard inflationary theory.

What about quantum mechanics makes the concept of a classic science-fiction teleportation device problematic?

Typically, a teleporter works by scanning the object you want to teleport, and then sending the information needed to build the object at the other end. This is a problem in the context of quantum mechanics because necessarily when you measure something, you also affect it, so you can't extract all the needed information about the quantum states of the particles you want to teleport without completely destroying those quantum states.

What are the basic steps of doing quantum teleportation for one particle, and how do they get around the issue from the previous question?

The basic procedure is as follows (for one particle):

1. Create a pair of entangled particles. Keep one and send the other to the location to which you want to teleport the particle.

2. Make a joint measurement of the particle you want to teleport and the entangled one you kept. Ensure that this measurement doesn't disrupt the particle to be teleported (i.e. it can't reveal the quantum state of the particle to be teleported, just how that state is related to the entangled particle's state).

3. Transmit the information from your joint measurement.

4. The person on the other end can use the transmitted information to apply the necessary manipulations to reproduce the quantum state of the original particle.

This avoids the usual difficulties with teleportation in a quantum world by never directly measuring the state of the particle being teleported. By avoiding learning about the state, you prevent the collapse of the wave function, thereby preserving the original quantum state.

1. Create a pair of entangled particles. Keep one and send the other to the location to which you want to teleport the particle.

2. Make a joint measurement of the particle you want to teleport and the entangled one you kept. Ensure that this measurement doesn't disrupt the particle to be teleported (i.e. it can't reveal the quantum state of the particle to be teleported, just how that state is related to the entangled particle's state).

3. Transmit the information from your joint measurement.

4. The person on the other end can use the transmitted information to apply the necessary manipulations to reproduce the quantum state of the original particle.

This avoids the usual difficulties with teleportation in a quantum world by never directly measuring the state of the particle being teleported. By avoiding learning about the state, you prevent the collapse of the wave function, thereby preserving the original quantum state.

If time travel is possible, what is the most promising approach to building a time machine based on our current understanding of general relativity?

The most promising approach would be use use a wormhole. If you created a wormhole, and then moved one end of the wormhole at speeds near the speed of light, time for the moving end would pass more slowly than for the stationary end. Once you were done moving, the two ends would be at different times (the moving end would be further into the future). Anyone passing from the moving end to the end that remained stationary would go back in time, while trips the other way would lead to the future.

Even if time travel is physically possible, why wouldn't we expect to see visitors from the future here today?

Based on the most likely scenario for time travel, using wormholes, it's impossible to travel backwards in time to before the wormhole was created. So no one from the future has visited us because we haven't invented a time machine yet.