Chapter 28 Lecture Flashcards

Hubble's Law

• Beginning of Modern View:
  • Before the 1920s, the Milky Way Galaxy was considered the entire universe.
  • Edwin Hubble disproved this in the early 1920s by discovering that the Andromeda "nebula" was a distant galaxy separate from our own.
• Hubble's Measure of Distance:
  • Hubble deduced the distance of a star by comparing its luminosity (energy) and brightness.
  • Brightness of light obeys an inverse square law: an object twice as far away appears one-quarter as bright.
• Hubble's Measure of Redshift:
  • Light from a source moving toward us is blueshifted (high frequency).
  • Light from a source moving away is redshifted (low frequency).
  • The faster the object's movement, the greater the shift.
• Observed Patterns:
  • Hubble found that most objects are moving away from each other.
  • The farther an object, the faster it moves away (greater redshift).
• Hubble's Equation:
  • v = H_0 \times d
  • v: velocity of the object.
  • H_0: Hubble's constant.
  • d: distance.
  • The equation shows that the velocity of an object is proportional to its distance.
• Implications:
  • Hubble's law indicates that the universe is expanding.
  • The expansion is analogous to an ant on an inflating balloon; the ant sees every point moving away.
  • It provides strong evidence for the Big Bang theory.

Our Observable Universe

• Limit:
  • The farthest we can see is about 14 billion light-years away.
  • This is due to observing light that has traveled to us for the entire age of the universe (approximately 14 billion years).
• The Observable Universe Back in Time:
  • When the observable universe was half, a quarter, 1/8, 1/16, and 1/32 of its current size, going backward in time, the squares representing space are shrinking because space itself is expanding.
  • This progression makes it appear as though the universe started from a point.
  • Important to remember that we are only considering the observable universe.
• More of the Universe Back in Time:
  • When we include the stuff outside our observable universe, even though the observable universe is still shrinking toward a point as we go back in time; look what is outside of it, the rest of the universe's material is shrinking around it.
  • The entire universe likely did not expand from a single point but was probably always infinite in size with no center to the expansion, with early Universe being extremely hot and dense.

The Big Bang

• Definition:
  • The theory that our universe began with a primordial explosion approximately 13.7 billion years ago.
  • Marks the beginning of space and time.
• Evidence:
  • Continuing expansion of the universe.
  • Measured cosmic background radiation, which was predicted before its discovery.
  • Measurements of element abundances, also predicted before measured.
• Cosmic Microwave Background (CMB):
  • If the Big Bang transpired and the universe is expanding, the light emitted during the Big Bang would be significantly redshifted, appearing as microwave signals today.
  • Arno Penzias and Robert Wilson detected these signals in 1964, termed the Cosmic Microwave Background.
  • The uniformity of the CMB suggests that all matter in the universe was once very close together.
  • The discovery of the CMB bolsters the Big Bang theory.
• Relative Abundance of Light Elements:
  • Elements form through the fusion of lighter elements, which requires high temperatures and densities.
  • Scientists posit that the rapid expansion following the Big Bang would have limited the formation of heavy elements, allowing only hydrogen and helium to form.
  • Measurements show that around 75% of the matter in the universe is hydrogen and about 25% is helium, which supports the Big Bang theory.

Cosmic Inflation

• Mysteries Needing Explanation:
  1. Where does the structure of the universe come from?
  2. Why is the overall distribution of matter so uniform?
  3. Why is the density of the universe so close to the critical density?
  • An early episode of rapid inflation can solve all three mysteries!
• Explanation of Key Features
  • Inflation can create all the structure by stretching tiny quantum ripples to an enormous size. These ripples in density then become the seeds for all structures in the universe.
  • Microwave temperature is nearly identical on opposite sides of the sky because regions now on opposite sides of the sky were close together before inflation pushed them far apart.
  • Inflation of the universe flattens its overall geometry like the inflation of a balloon, causing the overall density of matter plus energy to be very close to the critical density.
  • Overall geometry of the universe is closely related to total density of matter and energy.
  • Density = Critical: Flat Universe
  • Density > Critical: Spherical Universe.
  • Density < Critical: Saddle-Shaped Universe

General Relativity

• Definition:
  • Published by Einstein in 1915.
  • States that gravity is the consequence of mass distorting the fabric of spacetime.
• Reference Frames:
  • Einstein stated that observations inside an enclosed chamber cannot determine if the chamber is at rest or moving with constant velocity (1905).
  • Accelerated motion inside a chamber would be noticeable.
  • Einstein's belief was that the laws of nature should have the same form in every frame of reference, accelerated as well as non-accelerated, was the primary motivation that led him to the general theory of relativity.
• Principle of Equivalence:
  • Observations made in an accelerated reference frame are indistinguishable from observations made in a Newtonian gravitational field.
  • Einstein imagines a spaceship far from gravitational influences. At rest or in uniform motion, everything inside floats freely.
  • When the rocket motors are activated and the ship accelerates, gravity-like phenomena appear.
  • Dropping a wood ball and a lead ball inside a spaceship:
    • If the ship moves at a constant velocity, the balls remain suspended.
    • If the ship accelerates, the floor catches up with the balls simultaneously.
• Bending of Light by Gravity:
  • In a stationary spaceship in a gravity-free region, a ball thrown sideways follows a straight-line path relative to observers inside and outside the ship.
  • If the ship accelerates, an outside observer still sees a straight-line path, but an observer inside sees a curved path (parabola).
  • According to the principle of equivalence, if acceleration deflects light, so must gravity.
  • Einstein posited that gravity pulls on the energy of light because energy is equivalent to mass.
  • He predicted that starlight passing close to the Sun would be deflected by a measurable angle.
• Gravity and Time: Gravitational Red Shift:
  • Einstein’s theory posits that gravitation causes time to slow down.
  • Time runs slower in the direction of gravitational force.
  • A clock at Earth's surface runs slower than one farther away.
  • An atom on the Sun emits light of a lower frequency than the same element on Earth, an effect called gravitational red shift (lowering of frequency shifts the color toward the red). As a photon flies from the surface of a star, it loses energy (but not speed= and therefore frequency due to the star's gravity.
• Gravity and Space: Motion of Mercury:
  • Planets orbit the Sun in elliptical orbits, periodically moving closer and farther from the Sun.
  • Einstein found that the elliptical orbits of planets should precess due to varying gravitational fields, with the greatest precession near the Sun (Mercury).
• Gravity, Space, and a New Geometry:
  • Gravity causes space to be non-Euclidean and the laws of Euclidean geometry are invalid when applied to objects in strong gravitational fields.
  • The rules of Euclidean geometry are valid in flat space; but on a curved surface, like a sphere or a saddle-shaped object, the Euclidean rules no longer hold. The sum of the angles of a triangle depends on which kind of surface the triangle is drawn on.
  • Lines of shortest distance are called geodesic lines or simply geodesics. The path of a light beam follows a geodesic.
  • General relativity requires a new geometry where space is a flexible medium that can bend and twist.
  • Mass produces the curvature, or warping, of spacetime.
  • Masses respond in their motion to the warping of the spacetime they inhabit.
• Gravitational Waves:
  • Every object with mass warps the surrounding spacetime.
  • Changes in an object's motion cause the surrounding warp to move, producing ripples in the geometry of spacetime, known as gravitational waves.
  • These waves travel outward at the speed of light.
• Newtonian and Einsteinian Gravitation:
  • Einstein's theory showed that Newton’s law of gravitation is a special case of general relativity.
  • Newton’s law is still accurate for most interactions in the solar system.
  • Newtonian theory is used for computing trajectories of space probes to the Moon and planets.
  • Einsteinian physics is needed in cases like calculating Mercury’s precession or studying black holes.

Special Relativity

• Motion is Relative:
  • The place from which motion is observed and measured is a frame of reference.
  • An object may have different velocities relative to different frames of reference.
• Michelson interferometer:
  • A beam of light from a monochromatic source was separated into two beams with paths at right angles to each other; these were reflected and recombined to show whether there was any difference in average speed over the two back-and-forth paths.
  • The interferometer was set with one path parallel to the motion of Earth in its orbit.
  • Either Michelson or Morley carefully watched for any changes in average speed as the apparatus was rotated to put the other path parallel to the motion of Earth.
  • But no changes were observed.
• Postulates of Special Theory of Relativity:
  • All laws of nature are the same in all uniformly moving frames of reference.
  • The speed of light in free space has the same measured value for all observers, regardless of the motion of the source or the motion of the observer; that is, the speed of light is a constant.
• Simultaneity:
  • Two events are simultaneous if they occur at the same time.
  • Two events that are simultaneous in one frame of reference need not be simultaneous in a frame moving relative to the first frame.
• Spacetime:
  • Space and time are intimately linked together. Things exist in spacetime.
  • Each object, each person, each planet, each star, each galaxy exists in what physicists call “the spacetime continuum. ”
  • One observer’s measurements of space and time differ from the measurements of another observer in some other realm of spacetime in such a way that each observer will always measure the same ratio of space and time for light: the greater the measured distance in space, the greater the measured interval of time.
• Time Dilation:
  • Demonstrated by a light clock consisting of a flash of light bouncing between two parallel mirrors.
  • An observer moving with the clock sees the light flash moving vertically.
  • An observer seeing the moving clock observes the flash moving along a diagonal path.
  • The speed of light is the same in all reference frames, so the flash travels for a longer time between mirrors in the outside observer's frame.
  • t = \gamma t_0
  • Where:
    • t: relative time
    • t_0: proper time
    • \gamma: Lorentz factor
    • \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
  • Clocks tick slower as the speed increases, approaching the speed of light.
• The Twin Trip:
  • One twin takes a high-speed round-trip journey in the galaxy, while the other stays on Earth.
  • The traveling twin returns younger than the stay-at-home twin.
• Addition of Velocities:
  • For everyday objects:
    • V = v1 + v2
  • Relativistic rule for adding velocities:
    • V = \frac{v1 + v2}{1 + \frac{v1v2}{c^2}}
  • Light moving at c (speed of light) in one frame will be seen moving at c in any other frame.
• Length Contraction:
  • Space is contracted in the direction of motion, making objects look shorter at relativistic speeds.
    • L = L_0\sqrt{1 - \frac{v^2}{c^2}}
  • Where:
    • L: contracted length.
    • L_0: proper length.
• Relativistic Momentum:
  • p = \gamma mv
  • Where:
    • p: relativistic momentum.
    • \gamma: Lorentz factor.
    • m: mass.
    • v: velocity.
  • Subatomic particles pushed to near the speed of light behave as if their mass increases with speed.
• Mass, Energy and E = mc^2
  • A piece of matter, even at rest, has a "rest energy."
  • E = mc^2
  • Energy is related to mass by the equation above.
  • Mass and energy are equivalent.
• Correspondence Principle:
  • Any new theory must agree with the old where the old gives correct results.
  • Equations of special relativity must correspond to those of classical mechanics when speeds are much less than the speed of light.
  • When v << c, then \gamma = 1
  • Relativistic time: t = t_0
  • Relativistic length: L = L_0
  • Relativistic momentum: p = mv

Dark Matter

• Ordinary Matter:
  • Made of protons, neutrons, and electrons that form atoms.
  • Composed of elements listed in the periodic table.
• **What Makes Matter