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WEEK 4

FUNDAMENTALS OF PHYSICAL SCIENCE

THE INFINITESIMAL UNIVERSE

COURSE NOTES - PART 1

October 1927, Fifth Solvay International Conference on Electrons and Photons, where the world’s most notable physicists met to discuss the newly formulated quantum mechanics.

17 of the 29 attendees were or became Nobel Prize winners, including Marie Curie,, who alone among them, had won Nobel Prizes in two separate scientific disciplines.

Back to front, left to right: Back row: Auguste Piccard, Emile Henriot, Paul Ehrenfest, Edouard Herzen, Theophile de Dander, Erwin Schrodinger, JE Verschaffelt, Wolfgang Pauli, Werner Heisenberg, Ralph Fowler, Leon Brillouin; Middle row: Peter Debye, Martin Knudsen, William Lawrence Bragg, Hendrik Anthony Kramers, Paul Dirac, Arthur Compton, Louis de Broglie, Max Born, Niels Bohr; Front

Row: Irving Langmuir, Max Planck, Marie Curie, Hendrik Lorentz, Albert Einstein, Paul Langevin, 1 Charles-Eugene Guye, CTR Wilson, Owen Richardson

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Section 1: Quantum Mechanics

Quantum Mechanics is a mathematical formalism and area of study of physics which is mostly concerned (but not limited) with physical phenomena occurring in the micro world. “Micro” refers to atomic and subatomic scales 10-6 meters or less and at very low temperatures.

Why is Quantum Mechanics so important?

Quantum Mechanics emerged in the first three decades of the nineteenth century and has since become the driving force behind the electronic revolution. Very small matter like electrons do not follow the same laws of physics (i.e., gravity that governs the motion of planets and electromagnetism which describes interactions between charged particles).

Towards the end of the nineteenth century, a series of new discoveries related to the electronic structure of the atoms and molecules (atomic spectral lines, photoelectric effect) and to the nature of thermal radiation (blackbody radiation) could no longer be explained by the use of Classical Physics - Newton’s mechanics and Maxwell’s electromagnetic theory.

It was then necessary to develop a new theory that would satisfactorily explain these new phenomena, even if it meant to renounce some of the classical ideas. This new theory became known as quantum mechanics.

The first step was taken by Max Planck, who proposed on October 19th, 1900 that thermal radiation was emitted and absorbed not in a continuous way, but in discrete quanta in order to explain the spectra of thermal objects. He found the energy of the quanta was equal to the product of a constant:

Planck’s constant, h = 6.63 x 10-34 J·s

times the frequency of the mode (radiation).

In other words, the energy of the light (radiation) was quantized during emission and absorption.

The fact that hydrogen atoms emit or absorb radiation at a limited number of frequencies implies that these atoms can only absorb

Important Names: Albert Einstein:

“I have no special talent.

I am only passionately curious.”

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

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radiation with certain energies. This suggests that there are only a limited number of energy levels within the hydrogen atom. These energy levels are not continuous, but countable. The energy levels in any atom are quantized.

Another phenomenon that was puzzling physicists at the beginning of the twentieth century was the photoelectric effect: when ultraviolet light falls on a metal surface,

Light photons

electrons are ejected. This phenomena could not be explained by the common interpretation of light as an electromagnetic wave since the current produced did not depend on the intensity of incident light but only the frequency. The solution to the photoelectric effect was given in 1905 by Albert Einstein, who generalized Planck’s idea of quantization in such a way that these new quanta (photons) represent not only the process of absorption and emission of the light, but the light itself. For his work on the photoelectric effect, Einstein received the Nobel Prize in Physics in 1921.

Electrons ejected from the surface

Sodium metal

An excerpt from the website http://physics.info/photoelectric/, an analogy regarding

the photoelectric effect and how the voltage produced does not depend on the amplitude (intensity) of light, but only on frequency:

“The classical model of light describes it as a transverse, electromagnetic wave. Of this, there was very little doubt at the end of the Nineteenth Century. The wave nature of light was confirmed when it was applied successfully to explain such optical phenomena as diffraction, interference, polarization, reflection, and refraction. If we can imagine light as waves in an electromagnetic ocean and be quite successful at it, then it wouldn’t be much of a stretch for us to image electrons in a metal surface as something like tethered buoys floating in an electromagnetic harbor. Along came the waves (light) which pull and tug at the buoys (electrons). Weak waves have no effect, but strong ones just might yank a buoy from their mooring and set it adrift. A wave model of light would predict an energy-amplitude relationship and not the energy-frequency relationship

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described above. Photoelectric experiments describe an electromagnetic ocean where monstrous swells wouldn’t tip over a canoe, but tiny ripples would flight you into the air.

If that wasn’t enough, the photoelectrons seem to pop out of the surface too quickly. When light intensities are very low, the rate at which energy is delivered to the surface is downright sluggish. It should take a while for any one particular electron to capture enough of this diffuse energy to free itself. It should, but it doesn’t. The instant that light with an appropriate frequency of any intensity strikes a photoemissive surface, at least one electron will always pop out immediately (t = 10-9 s). Continuing with the ocean analogy, imagine a harbor full of small boats (electrons). The sea is calm except for tiny ripples on the surface (low intensity, short wavelength light). Most of the boats in the harbor are unaffected by these waves, but one is ripped from the harbor and sent flying upward like a jet aircraft. Something just ain’t right there. No mechanical waves behave like this, but light does.”

So far, we have seen that for a coherent explanation of some experiments, it is necessary to ascribe particle behavior to light.

The energy of such a particle (photon) of frequency, f, is: and its momentum:

wavelength of light

In 1922, American physicist, Arthur Holly Compton (1892-1962), provided convincing experimental evidence of the corpuscular (particle) nature of light.

Based on the fact that the photons associated with electromagnetic waves behave like particles, the French physicist, Louis de Broglie, proposed in 1923 that the particle and wave model should not be confined to forms of radiation but it should be applicable to all forms of motion, in particular, the motion of material particles (i.e., electrons). He suggested that all matter has wave-properties, completing the wave-particle duality. The wave-particle duality predicts that both the wave and the particle models apply to all objects whatever their size.

According to the wave-particle duality, material objects should have wave properties like interference and diffraction, so why don’t we see wave properties in material

objects all around us? When the energy and the momentum of the material particles is very high (> 100 MeV), their

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E hf

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wavelength is then so short (< 10-14 m) that they always behave like classical particles. For example, a small grain of sand with a mass of 10-9 kg moving at a speed of 10-3 m/s has a de Broglie wavelength of the order of 10-21 m and its wave properties are undetectable.

The state of the system is defined by the minimum amount of information (variables) which is sufficient to determine the conditions of the system.

For example, in Classical Physics, the state of the system at a given time is completely determined by the location (coordinates), orientation, and motion (momentum) of the system. In quantum mechanics, the specification of the variables that govern the state of the system is limited by a rule known as Heisenberg’s Uncertainty Principle that forbids us from simultaneously specifying (measuring) certain pairs of variables related to the system with arbitrary high precision.

An example of these variables are the location (coordinates) and the momentum of the particle. If we measure the position of the particle at a given instant, then the act of “measuring” would disturb the particle, leaving it in a state where its momentum has maximum uncertainty.

It is important to remember that this is not a statement about the inaccuracy of measurement devices or experimental procedures, but arises due to the inherent wave nature of quantum particles.

For example, by using DeBroglie’s equation, a sine wave of wavelength λ has a precisely defined momentum, while its possible positions are spread out all over space.

Please watch the following link for further information regarding experiments in wave particle duality: https://www.youtube.com/watch? v=DfPeprQ7oGc

Can we define a quantum state using variables that may be measured without destroying the established value of momentum? The answer is yes. For example, for a free electron (particle), it is possible to “find” the value of momentum (px) and the value of the energy (E) simultaneously. In such a case, the values of momentum (px) and energy (E) are the minimum amount of information (variables) which is sufficient to determine the state of the system.

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A fundamental postulate of quantum mechanics is that any system has a quantum state. This state is specified by the state function or wave function, ψ, which contain all possible information about the system. In our example, the state function is a function of the momentum and energy.

Because in quantum mechanics any measurement disturbs the system and generally leaves it in a different state, and because the output of measurements are generally uncertain, the most we can do is calculate the probabilities of

the different possible outcomes and compare them.

For example, while knowing that the particle is in state ψ(x), the most specific information about the position of the particle that could be obtained by a measurement is the probability of getting the position value, x, in a certain range.

Physical Interpretation of the State Function ψ

The square of the absolute value of the state function |ψ|2 is proportional to the probability of finding the value of the observable variable inside a certain range.

In classical physics, the time evolution of the state of the system is given by Newton’s laws of motion. In quantum mechanics, the change of state is given by the state function ψ and its evolution equation, the Schrodinger equation.

@ ~2 @2 i~@t (x,t)= 2m@x2 (x,t)+V(x) (x,t)

Summary:

· At the end of the twentieth century, a series of experimental observation (black body radiation, photoelectric effect, atoms’ spectral emissions) could not be explained by the use of classical physics.

· To explain the observations, Max Planck (1900) and Albert Einstein (1905) respectively introduced the idea of quantization. Light (radiation) is transmitted, absorbed, and emitted in the form of discrete units, new particles called photons. Quantum mechanics was born.

· By 1924, Louis de Broglie proposed that similar to light (radiation), which has particle-like properties, matter also has wave-like properties. The wave-particle dualism was extended to matter.

· The wave-particle duality of quantum mechanics predicts that both the wave and the particle models apply to all objects whatever their size. We don’t observe wave properties in common size objects because their wavelength is extremely small.

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• In quantum mechanics, a rule (Heisenburg’s uncertainty principle) forbids us from measuring certain pairs of observable physical quantities, like position and momentum with undefined precision.

• In quantum mechanics, the state of the system is described with the use of the state function ψ, which holds all of the information that is available about the system and its absolute square is proportional to the probability of finding the value of the observable variable in certain range.

• The time evolution of the state is given by the wave function and the Schrodinger equation.

Section 2: Blackbody Radiation

Radiation is the process by which energy is transmitted by means of electromagnetic waves.

Blackbody radiation is the electromagnetic radiation by objects in thermal equilibrium with its environment (at nonzero absolute temperature), at this point the amount of energy absorbed by the object is equal to the amount of energy being emitted by the object, otherwise the object’s temperature would change and the object would not be in equilibrium.

Because in equilibrium, emission and absorption are balanced then a material, which is a good absorber, is also a good emitter and a material, which is a poor absorber, would be a poor emitter. A perfect blackbody (and ideal body), is a perfect absorber and hence a perfect emitter.

It could be approximated by a small opening into a heating cavity (absorbs 98% of the incident radiation).

The Sun’s surface temperature is approximately 6000 K.

All objects emit radiation and for real-world objects, the radiation can be approximated by blackbody radiation. For example: the principles of blackbody radiation can be applied to the Sun, planets, stars, climates changes, an incandescent light bulb, and the kitchen stove.

The study of the data from blackbody emitters led to the formulation of Wien’s Displacement Law which relates the temperature of the object and the wavelength of maximum energy density.

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Notice that the hotter the object the bluer the radiation.

When the temperature increases, the peak shifts to lower wavelengths.

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THE MASSIVE UNIVERSE

COURSE NOTES - PART 2

Section 3: The Evolution of Stars

Before we talk about stars, we need to talk about the space in between stars. The density of this space is incredibly low - approximately a trillion trillion times less dense than matter in either stars or planets, and far thinner than even the best vacuum attainable in laboratory conditions on Earth. So why would we even discuss this near-perfect vacuum?

1. All added together, there is as much mass in between the stars as there is in the stars themselves.

2. Thisistheregionofspaceoutofwhichnewstarsareborn,andtheregionofspacewhereoldstars expel their matter when they die.

The matter in between stars is called interstellar matter and interstellar matter collectively is known as the interstellar medium. It consists of two main components: gas and dust. The gas is made up of individual atoms and small molecules, while the dust consists of clumps of atoms or molecules. When there is an area of denser-than-average gas and dust, we call that an interstellar cloud.

Star formation happens when part of an interstellar cloud collapses. Clouds (not unlike stars, as we will learn soon) must maintain a type of equilibrium to keep from expanding (by internal heat) or collapsing (by gravity). When something happens that disrupts that equilibrium, like a shock wave from a nearby supernova, the gravitational force will take over and the cloud will collapse.

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When considering just a few atoms, the gravitational force is nowhere near strong enough to overcome the random thermal motion (heat) of the atoms. The atoms will come near each other, temporarily clump together, but then disperse. However, as the number of atoms increases, the total mass of the cloud increases, and the gravitational force increases as well. Once the number of atoms reaches about 1057 (which coincidentally, is around the number of atoms in our Sun), then the atoms will no longer disperse.

As stated above, 1057 atoms do not generally clump together by chance and are instead triggered when a sufficiently massive amount of gas is squeezed together by some external event. This could be a shock wave from a nearby supernova, the death of of a massive star, or a shock wave produced when a nearby group of stars form and heat their surroundings in creation of an emission nebula. Regardless of the cause, theory suggests that once that collapse is triggered, star formation is inevitable.

The interstellar cloud begins to contract, likely triggered by a shock wave or a

pressure wave from a nearby catastrophic event (like a supernova explosion). As the cloud contracts, it fragments into smaller pieces due to gravitational instabilities in the gas. A typical cloud can break up into hundreds or thousands of fragments.

Stage 1:

Stage 2:

Individual cloud fragments begin to collapse. Once the density is high enough, there is no further

fragmentation. This occurs because as the density increases, radiation can no longer escape. The trapped radiation causes the temperature to rise, the pressure to increase, and the fragmentation to stop. However, the contraction continues.

Stage 3:

The Stage 2 fragment has now shrunk into a gaseous sphere with a diameter roughly the size of our Solar System. The inner regions have become opaque to their own radiation and have heated up considerably. For the first time, the fragment is beginning to resemble a star.

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Protostars occur in the stage of star formation when the interior of a collapsing fragment of gas is sufficiently hot and dense that it becomes opaque to its own radiation. The protostar is the dense region at the center of the fragment.

The Orion Nebula is thought to contain interstellar clouds in the process of condensing, as well as protostars.

Stage 4:

As the protostar evolves, its density increases and the temperature rises, both in the core and on the surface (called the photosphere). When the core of the star reaches 1 million Kelvin (K), electrons and protons are ripped from atoms and move at speeds of hundreds of kilometers per second. The protostar is still much larger than the Sun, now about the size of Mercury’s orbit.

At this point in a star’s evolution, its physical properties can be plotted on an important chart called the H-R Diagram. The H-R Diagram is a scatter plot of stars showing the relationship between the stars’ luminosities against their surface temperatures. At each phase of a star’s evolution, this can be represented by a single point on the diagram. The motion of that point around the diagram as the star evolves is called the star’s evolutionary track. It is a graphical representation of a star’s life and does not have anything to do with the actual spatial motion of the star.

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Stages 5, 6, and 7:

Internal heat is continuously increasing and the outward pressure created by that heat is starting to counteract and slow the gravitational contraction. The balance is not yet perfect, so the contraction slows, but does not stop completely. Planetary formation can begin in these final stages. The star will eventually heat up so much that it reaches the magic number of 10 million K, where a process known as nuclear fusion can begin. Here, the protostar has become a star. The star continues its contraction and increasing in temperature until it reaches a state of equilibrium. When it reaches equilibrium, it is said to be on the main sequence and it will remain there for most of its life.

It takes a star a few tens of millions years to reach the main sequence, but will live on the main sequence for about 10 billion years before evolving into something else. The end of this section will focus on that “something else” but first let us review the requirements for a star on the main sequence. There are three main components:

1. A star must be in hydrostatic equilibrium. This is the balance between the internal thermal pressure (heat) pushing outward and the weight of the material pressing inward toward the center of the star due to gravity. This is what allows the star to maintain a constant size.

2. A star must also be in thermal equilibrium. This means that the energy production at the core of the star equals the energy radiated off the surface. In other words, the star shines at a constant rate.

3. Thestar’smainsourceofenergymustcomefromthenuclearfusionofhydrogenintoheliumat the core.

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The full process in the proton-proton chain.

One of the major processes of fusing hydrogen into helium is known as the proton-proton chain. The specific mechanics of this are shown in the graphic to the right, but are not super important for our discussions in this course. The main thing we need to know is that hydrogen atoms are smashed together at really high speeds to ultimately form helium. The resulting helium atom has less mass than the original mass of the hydrogen atoms, so in order for mass to be conserved throughout the process, some of that mass is given off in the form of energy. This process is the fuel of all main- sequence stars.

A natural question might have come to you at this point - if stars are turning all their hydrogen into helium in their cores, won’t they eventually run out of hydrogen? The answer is yes. There is a finite supply of hydrogen in the core, so eventually they will run out and this is the reason that stars cannot and do not live forever.

So we know what happens to the hydrogen in the core, but what about the helium? What happens to the helium after it is formed? The answer at this point is “not a whole lot”. The temperatures in the core are not hot enough for helium to begin fusing, so it forms and then sinks to the center of the core doing nothing, kind of like ashes forming in the bottom of a fireplace. This pile of currently useless helium builds up slowly as more and more hydrogen fuses together. Eventually, much later in the star’s lifetime, when a large percentage of the hydrogen has been converted into helium, the core has gained mass due to its increasingly heavy pile of helium and begins to gain its own appreciable gravitational pull. This is important, because when all of the hydrogen is eventually converted into helium, the nuclear reactions stop and gravity takes over, causing the core to shrink.

Stage 8:

When the core collapses, the layers outside of the core begin to collapse as well. As they collapse, they heat up significantly, and eventually will heat up so much that the nuclear fusion of hydrogen (hydrogen burning) can begin in the layer just outside of the core. This process is called shell burning and brings our star out of the main sequence into the subgiant phase So we now have three layers to our star: a collapsing, non-fusing helium “ash” of a core, a hydrogen- burning shell immediately outside of that, and a non-burning envelope of gas surrounding them both.

Stage 9:

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Hydrogen burning continues in a shell around the core, producing more energy than needed to support the star at its current size. This means that the star has left its state of hydrostatic equilibrium, the internal pressure pushing outward has increased, and the outer envelope will begin to expand.

Now we have two separate things happening inside the star - the core is contracting and the outer shell is expanding. The star has now become a red giant, and its radius would extend out as far as the orbit of Mercury if it was in the Sun’s current position.

It gets a little bit more complicated when the core

contracts to a certain density. At this point, the core is dense enough that the pressure of the electron shells will

not allow the core to contract anymore. So it will not get any smaller (or larger), the pressure will not increase, but the temperature will continue to climb. The core is now acting more like a liquid than a gas, in that it is able to heat up without expanding. In addition to reaching this point, remember that the layer just outside of the inner core is fusing hydrogen into helium, and the helium is sinking toward the center onto the core just like it did on the main-sequence. The addition of the helium is increasing the mass and temperature of the core.

Stage 10:

Eventually, the core reaches a temperature of 100 million Kelvin (K), which is hot enough for helium fusion to begin. Helium fusion requires higher temperatures than hydrogen fusion, because it is heavier and requires greater speeds for the atoms to smash together in order to combine. Helium fusion is called the triple-alpha process, because it converts three helium nuclei into one carbon atom.

The triple-alpha process is highly temperature dependent: doubling the temperature of the reaction will cause it to run roughly a trillion times faster. As the fusing helium heats the core (which can still not expand due to its previous state), the increased temperature causes the helium fusion to suddenly proceed millions of times faster, which very quickly heats the core even more, which in turn causes the helium to fuse much, much faster.

In short, the center of the helium core explodes. About 6% of the core is fused into carbon within a few minutes. This corresponds to burning roughly 10 Earth masses of helium per second. This is called the helium flash. Within just a few minutes, the helium flash releases as much energy as our current Sun generates in 200 million years! Within hours, the enormous output of energy is over, and the star once again reaches a state of equilibrium.

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Stages 11 and 12

As the remaining helium fuses into carbon, a new carbon-rich inner core begins to form in a phenomena similar to the earlier build-up of helium. Helium becomes depleted at the center, and eventually fusion stops. The same process as when hydrogen was depleted starts again - the core shrinks and heats up, the outer envelope expands, and the star becomes a (larger) red giant.

If a star has enough mass, the core can heat up high enough for carbon fusion to occur, still heavier elements could be created, and the newly generated energy might again support the star and restore equilibrium.

For stars like our Sun, however, this is where it ends. Our Sun does not have enough mass to get the temperature high enough to fuse anything past helium.

Section 4: The Death of Stars

After helium fusion ceases, there is no more outward pressure being generated in the core of low- mass stars. Gravity will take over with nothing to fight against it and the core will contract. The carbon core, for all practical purposes, is dead, while the outer shells use up their hydrogen and

helium fuel at an increasing rate. The outer shell expands, cools, and then the star begins to fall apart. The outer layers begin to drift away into interstellar space, forming a planetary nebula.

Once the nebula is gone, the remaining core is extremely hot, dense, and small and is known as a white dwarf. With nothing to maintain the temperature of the white dwarf star, it cools and gets dimmer and dimmer, and eventually stops radiating light all together.

White Dwarf

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It can be seen from the H-R diagram that the evolutionary track of more massive stars once they leave the main sequence is much different than stars around the same size as our Sun.

SUN-SIZED STARS MASSIVE STARS

Stars with more mass (more than 10 times more massive than the Sun) can continue fusing heavier and heavier elements until they get to iron. Once the inner core of a star begins to change to iron, our star is in trouble. Nuclear fusion involving iron does not produce extra energy, because iron nuclei are so tightly bound that energy cannot be extracted by combining them into heavier elements. So the foundation of the star is destroyed, and its equilibrium is gone forever. Even though the temperature is incredibly hot at this point (up to several billion K), the enormous inward gravitational pull of matter is building up to a catastrophic end for a our high-mass star. Gravity will overwhelm the pressure of the hot gas and the star will implode, falling in on itself rapidly.

The speed at which this collapse occurs in remarkable, about a quarter of the speed of light. It takes a single second for the core to go from the size of the Earth to about the size of Ohio! Eventually the density of the core reaches just under the density of a neutron, about 1014 grams per cubic centimeter. At this point, there is an outward pressure that tries to stop the collapse that comes from something called nuclear binding energy. Nuclear binding energy usually works to keep the nucleus of an atom together, but it can also act in the other direction, as it does in this case. The nuclear binding energy tries to stop the collapse, but it’s happing so quickly that it overshoots, it overcompensates, and it sends the energy back out in the opposite direction at about half the speed of light. The core has essentially rebounded from its collapse and is now flying back outward (picture a slow motion video of a bouncy ball colliding with a wall). This rebound mechanism has so much energy that it blasts all the remaining material in the star out into space in the form of a supernova explosion.

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The Crab Nebula is a remnant from a supernova explosion that occurred in the year 1054.

In summary, low-mass stars die quietly while high- mass stars die with a bang. Star formation is a cyclical process. Stars form, evolve, and die. In dying, they send heavier elements into the interstellar medium, and those elements become parts of new stars.

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THE INFINITESIMAL AND THE MASSIVE

COURSE NOTES - PART 3

Section 5: Star Remnants

What is left after a supernova explosion?

The iron core of the massive star collapsed until its neutrons effectively came in contact with one another. The central portion of the core rebounded, creating a powerful shock wave that violently expels matter into space. This shock wave does not start at the very center of the collapsing core. The innermost part of the core - the part that rebounded - remains intact while the shock wave it created destroys the rest of the star. After the supernova, this portion of the core remains as a super dense ball of neutrons and we call this a neutron star.

Neutron stars are incredibly small and incredibly massive. A typical neutron star is not much bigger than a small asteroid or a city on Earth. Its mass, however, is greater than the Sun.

Fun Fact: A teaspoon full of neutron star would weigh as much as all of humanity (about 7 billion people)!

A typical neutron star compared to the size of Manhattan

Neutron stars are solid objects. You could even imagine standing on it, except for the fact that the gravity is so powerful that a 150 pound person would weigh the Earth equivalent of 1 million tons on a neutron star. The severe pull of a neutron star’s gravity would instantly flatten you into a pile of people jelly about one atom thick.

In addition to being very massive and very small, neutron stars have a few other important properties. They rotate very quickly, with periods measured in just fractions of a second. They also have incredibly strong magnetic fields, trillions of times stronger than Earth’s.

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Neutron stars are supported by the resistance of tightly packed neutrons to further compression. Squeezed together, these particles form a hard ball of dense matter that it does not seem that even gravity should be able to compress it further. But it turns out, given enough mass packed into a small enough volume, the collective pull of gravitational forces can crush even a neutron star. The neutron star shrinks to the size of a planet, to an asteroid, to the tip of a needle, then even smaller than that.

This is a black hole. The gravitational pull in the vicinity of a black hole becomes so great that nothing, not even light, can escape. Therefore, a black hole will emit no light, no other form of radiation, no information that we can perceive whatsoever.

Newtonian (classical) mechanics has been the tool that we have used to study the universe up until this week. However, its laws break down in or near black holes. It cannot adequately describe the conditions there. We have to use a more modern theory of gravity, Einstein’s general theory of relativity, which we will discuss shortly.

Still, we can usefully describe some aspects of black holes in more or less Newtonian terms. For example, according to classical physics, the escape speed of an

object is the speed required for one object to reach in order to escape from the gravitational pull of another object. The escape speed of an object to escape the gravitational pull of Earth is 11,200 m/s. The escape speed for an object to escape the gravitational pull of the Sun is 42,500 m/s. This is calculated by:

v = escape speed (m/s) G = gravitational constant = 6.67 x 10-11 Nm2/kg2 M = mass (kg) R = radius (m)

Considering this equation, let’s consider compressing the mass of the Earth down to a quarter of its current size. The mass would stay the same, but the radius would decrease, so the escape speed would go up. If we continue to compress the Earth down to about the size of a penny, the speed needed to escape its surface would reach 300,000,000 m/s - the speed of light.

Astronomers have a special name for the critical radius at which the escape speed from an object would equal the speed of light and within which the object could no longer be seen. This is called the Schwarzschild radius. The surface of the imaginary sphere with a radius equal to the Schwarzschild radius is called the event horizon, sometimes referred to as the “surface” of the black hole.

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2GM R

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Any object can become a black hole. The Schwarzschild radius is the radius to which the object would have to be compressed in order to become a black hole. This is directly proportional to an object’s mass - the more massive the object, the greater the size of the Schwarzschild radius. For example, Earth would need to be compressed to a radius of 1 centimeter. Jupiter is much more massive than Earth, so it would only need to be compressed to a radius of 3 meters. The Sun is much more massive than Jupiter, so it would need to be compressed to a radius of 3000 meters.

So if we cannot see black holes, how do we know that they exist? They do not reflect or emit light, but their presence can be detected through the effects of their gravitational fields on nearby objects.

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Section 6: Relativity

We learned at the very beginning of this course about Isaac Newton’s contributions to the physical laws of motion. One of his other great contributions was his Universal Law of Gravitation. This law states that all objects that have mass in the universe are attracted to each other. The more massive the objects, the greater the gravitational force between them. The closer the two objects are to one another, the greater the gravitational force between them.

Newton’s law of gravitation first appeared in print in the year 1687 and was considered a simple, yet powerful equation. It can be used to predict the positions of all celestial objects and also to calculate how much energy is needed to break through the gravitational bonds of Earth. Every astronaut and every satellite that ever went into space began their journey with this equation.

This law (and Newton’s laws of motion) remained largely unquestioned until the beginning of the 1900s when Albert Einstein shook the foundations of physics with his introduction of his theories of relativity. His Special Theory of Relativity showed that Newton’s laws of motion were only approximately correct and begin to break down at velocities that approach the speed of light. His General Theory of Relativity showed that Newton’s law of gravitation was also only approximately correct and would break down in the presence of very strong gravitational fields.

Einstein introduced special relativity in 1905, and general relativity in 1915. We will address the two in the same order.

Special Relativity, Part I

The speed of light is the maximum possible speed, and it is always measured to have the same value by all observers.

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Special Relativity, Part II

A bullet fired from a speeding car will be measured by an outside observer to have a velocity equal to the sum of the velocities of the car and the bullet.

However, a beam of light shining forward from a moving spacecraft will still be measured by an outside observer to have a velocity of c (300,000,000 m/s) regardless of the speed of the spacecraft. The speed of light is independent of of the speed of the source or the speed of the observer.

There is no absolute frame of reference and no absolute frame of rest.

This means that there is no “preferred” observer relative to whom all other velocities can be measured. We may feel like we are preferred observers on Earth, but as you “stand still” on Earth’s surface, the Earth is rotating at a speed of 1000 mph, it is orbiting around the Sun at a speed of 67,000 mph, the Sun is orbiting around the center of the Milky Way at a speed of 450,000 mph, and the Milky Way is moving through intergalactic space at a speed of over a million mph. So when we “stand still” on Earth, we are moving well over a million mph as well! Therefore, velocities measured from our location in space are relative. Only relative velocities between observers matter.

Special Relativity, Part III Space and time are not independent, but can instead be identified as “spacetime”.

Neither space nor time can be considered independently of each other in Einstein’s relativity. They are no components of a single entity known as spacetime. So not only is there no absolute frame of reference in the universe, there is also no absolute, universal time. Depending on motion, observer’s clocks will tick at different rates and their time will measure differently.

It is important to remember that special relativity and Newtonian mechanics are equivalent when describing objects that are moving much more slowly than the speed of light, but measurements and calculation will differ greatly at speeds approaching the speed of light (relativistic speeds).

Einstein commonly used thought experiments to address questions in his theories of relativity, because physical experiments are largely impractical to perform because of the extreme conditions

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required. Thought experiments allow us to see how the concepts of space and time must be altered in order to accept Einstein’s theories.

We will now conduct a few thought experiments to address one of the consequences of Einstein’s special relativity, a phenomena known as time dilation.

Thought Experiment #1

Q: Sally is standing in a field and John is riding on a railroad car that is moving with velocity, v. John shines a flashlight in the direction in which he is moving. What do they both observe?

A: Because of Part 1 of special relativity, the principle of the constancy of light, we know that each observer will measure the beam of light from the flashlight as traveling at the same speed.

Thought Experiment #2

Q: Sally is standing in a field. Next to her is a light clock. That is, two mirrors that are reflecting a beam of light back and forth, and the journey from one mirror to the other and back again counts as one tick of the clock. Also, Sally is wearing a watch that is synchronized with her light clock.

John is standing on a stationary railroad car. He also has a light clock, and his clock is synchronized with Sally’s and his own wristwatch. What happens?

A: Nothing unusual happens. John’s watch and clock stay perfectly synchronized with each other and with Sally’s.

Thought Experiment #3

Q: We have the same setup except the railroad car is now moving to the left with velocity, v. What happens?

A: From John’s perspective, the beam of light keeps going up and down between the mirrors, but from Sally’s perspective, the light now has to travel a diagonal path from one mirror to another. Since Sally still measures the speed of light as c, she is now going to observe John’s light clock as ticking slower than hers since the light now has a longer distance to travel. However, since John is moving along at the speed as the railroad car, he still experiences his watch as being synchronized with his light clock, so Sally will see his watch slow down along with his light clock!

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Time dilation is the difference of elapsed time between two events as measured by observers moving relative to each other. It can be calculated by:

t’ = dilated time (s) t = stationary time (s) v = velocity (m/s) c = speed of light (m/s)

The closer an object gets to the speed of light, the greater the difference in measured times.

For example, for an object moving at 10% the speed of light:

t 1−0.01c2

t 1−(0.1c)2

t c2

t

t 1−0.01

t

1−v2 c2

c2 t

The dilated time will be 1.0005 times greater than the stationary time.

tt 0.9949874371

t1.005t

t 0.99

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For an object moving at 95% the speed of light:

t 0.9025c2

t 1−(0.95c)2

t c2

t

t 1 −0.9.025

tt 0.31224989992

t3.2t

The dilated time will be 3.2 times greater than the stationary time.

1−

t 0.0975

In constructing the special theory of relativity, Einstein rewrote the laws of motion set forth by Isaac Newton centuries before him. Fitting Newton’s other great legacy, the theory of gravitation into the framework of relativity was a much more complex mathematical problem that took Einstein another decade to solve. The problem was that special relativity is only valid for systems that are not accelerating. Since we know from Newton’s second law of motion that acceleration implies a force, this means that special relativity is not valid when a force is present. Therefore, it cannot generally be used when there is a gravitational Oield present. The general theory of relativity was Einstein’s solution to removing that restriction on special relativity.

General relativity used the equivalence principle to reason that there is no way to tell the difference between a gravitational Oield and an accelerated frame of reference.

Therefore, Einstein could address gravity as simply an acceleration of particles, but in order to do so, spacetime needed to be curved.

The central concept of general relativity is that matter - all matter- tends to “warp” or curve the space in its vicinity. Objects such as planets and stars react to this warping by changing their paths.

In the Newtonian version of gravity, particles

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move on curved paths because they are acted on by a gravitational force. In Einstein’s version, those same particles move on curved paths because they are free falling through space, simply following the curvature of spacetime produced by a massive nearby object.

Matter tells space how to curve, and space tells matter how to move.

So matter warps space, but remember that we no longer consider space and time independent quantities of each other. Therefore, we can say that matter warps spacetime and in doing so, it redeOines straight lines (i.e., the path a beam of light would take). The more massive an object is, the more it warps space and time around it.

A black hole occurs when the “hole” produced by matter warping spacetime becomes inOinitely deep.

Section 7: Dark Matter, Dark Energy, and the Big Bang

Dark matter is a recently discovered new state of matter that surrounds galaxies and is not yet well understood. Dark matter does not interact with electromagnetic waves, so it cannot be seen or touched. Similar to black holes, the only reason we know that it exists is through its gravitational effects on other masses.

Dark matter was discovered when the stars at the edge of galaxies were found to be traveling much faster than predicted. Once all of a galaxy is within an orbit, the velocity should diminish with distance, as the dashed curve labeled “Keplerian motion” indicates below. However, even in our own Milky Way galaxy, it doesn’t show as predicted. For the rotation speeds measured of gases outside of the visible portion of the galaxy to produce the observed curve, more than twice the mass of the galaxy would have to be outside the visible part.

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Galaxy mass measurements show that galaxies need between 3 and 10 times more mass than can be observed to explain their rotation curves. The discrepancy is even larger in galaxy clusters, which need 10 to 100 times more mass!

Dark energy is a recently discovered state of energy existing in empty space and is not yet well understood. Dark energy was discovered when the universe was found not only to be expanding, but also that the rate of expansion was accelerating.

Matter curves the fabric of space inward, while dark energy curves the fabric of space outward.

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The Big Bang model of the universe states that all matter and energy erupted from an inOinitesimal point and expanded into its current state.

The future of our universe is unknown, but with the discovery of dark energy, scientists speculate that the universe will continue to expand without halting.