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Explore the principles of General Relativity, including its extension from Special Relativity, tests and experiments that confirm its predictions, the phenomenon of gravitational waves, and the formation of black holes and neutron stars.
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CHAPTER 15General Relativity • 15.1 Tenets of General Relativity • 15.2 Tests of General Relativity • 15.3 Gravitational Waves • 15.4 Black Holes and Neutron Stars Time and space and gravitation have no separate existence from matter. Albert Einstein
15.1: Tenets of General Relativity General relativity is the extension of special relativity. It includes the effects of accelerating objects and their mass on spacetime. As a result, the theory is an explanation of gravity. It is based on two concepts: (1) the principle of equivalence, which is an extension of Einstein’s first postulate of special relativity and (2) the curvature of spacetime due to gravity.
15.2: Tests of General Relativity Bending of Light • During a solar eclipse of the sun by the moon, most of the sun’s light is blocked on Earth, which afforded the opportunity to view starlight passing close to the sun in 1919. The starlight was bent as it passed near the sun which caused the star to appear displaced. • Einstein’s general theory predicted a deflection of 1.75 seconds of arc, and the two measurements found 1.98 ± 0.16 and 1.61 ± 0.40 seconds. • Since the eclipse of 1919, many experiments, using both starlight and radio waves from quasars, have confirmed Einstein’s predictions about the bending of light with increasingly good accuracy.
Gravitational Lensing • When light from a distant object like a quasar passes by a nearby galaxy on its way to us on Earth, the light can be bent multiple times as it passes in different directions around the galaxy.
Light Retardation • As light passes by a massive object, the path taken by the light is longer because of the spacetime curvature. • The longer path causes a time delay for a light pulse traveling close to the sun. • This effect was measured by sending a radar wave to Venus, where it was reflected back to Earth. The position of Venus had to be in the “superior conjunction” position on the other side of the sun from the Earth. The signal passed near the sun and experienced a time delay of about 200 microseconds. This was in excellent agreement with the general theory.
15.4: Black Holes • While a star is burning, the heat produced by the thermonuclear reactions pushes out the star’s matter and balances the force of gravity. When the star’s fuel is depleted, no heat is left to counteract the force of gravity, which becomes dominant. The star’s mass collapses into an incredibly dense ball that could warp spacetime enough to not allow light to escape. The point at the center is called a singularity. • A collapsing star greater than 3 solar masses will distort spacetime in this way to create a black hole. • Karl Schwarzschild determined the radius of a black hole now known as the Schwarzschild radius.
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole
15.3: Gravitational Waves • When a charge accelerates, the electric field surrounding the charge redistributes itself. This change in the electric field produces an electromagnetic wave, which is easily detected. In much the same way, an accelerated mass should also produce gravitational waves. • Gravitational waves carry energy and momentum, travel at the speed of light, and are characterized by frequency and wavelength. • As gravitational waves pass through spacetime, they cause small ripples. The stretching and shrinking is on the order of 1 part in 1021 even due to a strong gravitational wave source. • Due to their small magnitude, gravitational waves are difficult to detect. Large astronomical events create measurable spacetime waves such as the collapse of a neutron star, a black hole or the Big Bang. • This effect has been likened to noticing a single grain of sand added to all the beaches of Long Island, New York. blob:https://www.newyorker.com/117ea397-715a-4022-8dc1-1db0c17623e7
Gravitational Wave Experiments • Taylor and Hulse discovered a binary system of two neutron stars that lose energy due to gravitational waves that agrees with the predictions of general relativity. • LIGO is a large Michelson interferometer device that uses four test masses on two arms of the interferometer. The device is meant to detect changes in length of the arms due to a passing wave. • NASA and the European Space Agency (ESA) were jointly developing a space-based probe called the Laser Interferometer Space Antenna (LISA) which was to measure fluctuations when a gravitational wave passes.
GRAVITATIONAL WAVES DETECTED FROM COLLIDING NEUTRON STARS • emit • Emit both gravitational waves and light
Across what potential difference has an electron be accelerated to reach a speed of v=1.8^7m/s?
CHAPTER 3The Experimental Basis of Quantum Physics • 3.1 Discovery of the X Ray and the Electron • 3.2 Determination of Electron Charge • 3.3 Line Spectra • 3.4 Quantization • 3.5 Blackbody Radiation • 3.6 Photoelectric Effect • 3.7 X-Ray Production • 3.9 Pair Production and Annihilation
3.1: Discovery of the X Ray and the Electron X rays were discovered by Wilhelm Röntgen in 1895. Observed x rays emitted by cathode rays bombarding glass Electrons were discovered by J. J. Thomson. Observed that cathode rays were charged particles
Cathode Ray Experiments In the 1890s scientists and engineers were familiar with “cathode rays”. These rays were generated from one of the metal plates in an evacuated tube across which a large electric potential had been established. It was surmised that cathode rays had something to do with atoms. It was known that cathode rays could penetrate matter and their properties were under intense investigation during the 1890s.
Observation of X Rays Wilhelm Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. These rays were unaffected by magnetic fields and penetrated materials more than cathode rays. He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube.
Röntgen’s X Ray Tube • Röntgen constructed an x-ray tube by allowing cathode rays to impact the glass wall of the tube and produced x rays. He used x rays to image the bones of a hand on a phosphorescent screen.
Apparatus of Thomson’s Cathode-Ray Experiment • Thomson used an evacuated cathode-ray tube to show that the cathode rays were negatively charged particles (electrons) by deflecting them in electric and magnetic fields.
12. How did Thomson measure the charge to mass ratio of the electron? a.He shot helium nuclei into gold foil to measure the nuclei scattering against electrons within. b.He passed cathode rays through a magnetic field and measured the deflection. c.He suspended a drop of oil between electrodes to measure the electric field from the electrons. d. He measured very precisely a known quantity of hydrogen atoms and calculated the reduced mass ratio within each atom.
Thomson’s Experiment • Thomson’s method of measuring the ratio of the electron’s charge to mass was to send electrons through a region containing a magnetic field perpendicular to an electric field.
Calculation of e/m • An electron moving through the electric field is accelerated by a force: • Electron angle of deflection: • The magnetic field deflects the electron against the electric field force. • The magnetic field is adjusted until the net force is zero. • Charge to mass ratio:
3.2: Determination of Electron Charge Millikan oil drop experiment
Oil drop is in motion, either falling without E, or rising with E
Calculation of the oil drop charge(at rest) • Used an electric field and gravity to suspend a charged oil drop • Magnitude of the charge on the oil drop • Mass is determined from Stokes’s relationship of the terminal velocity to the radius and density • Thousands of experiments showed that there is a basic quantized electron charge C
3.3: Line Spectra • Chemical elements were observed to produce unique wavelengths of light when burned or excited in an electrical discharge. • Collimated light is passed through a diffraction grating with thousands of ruling lines per centimeter. • The diffracted light is separated at an angle q according to its wavelength λ by the equation: where d is the distance between rulings and n is an integer called the order number
Optical Spectrometer • Diffraction creates a line spectrum pattern of light bands and dark areas on the screen. • Wavelengths of these line spectra allow identification of the chemical elements and the composition of materials. • a • l • lo.
Balmer Series • In 1885, Johann Balmer found an empirical formula for wavelength of the visible hydrogen line spectra in nm: nm (where k = 3,4,5… and k > 2)
Rydberg Equation • As more scientists discovered emission lines at infrared and ultraviolet wavelengths, the Balmer series equation was extended to the Rydberg equation: (n =1, 2,3…)
Transitions in the Hydrogen Atom Lyman series The atom will remain in the excited state for a short time before emitting a photon and returning to a lower stationary state. All hydrogen atoms exist in n = 1 (invisible). Balmer series When sunlight passes through the atmosphere, hydrogen atoms in water vapor absorb the wavelengths (visible).
Current theories predict that charges are quantized in units (quarks) of ±e/3 and ±2e/3, but quarks are not directly observed experimentally. The charges of particles that have been directly observed are quantized in units of ±e. The measured atomic weights are not continuous—they have only discrete values, which are close to integral multiples of a unit mass. 3.4: Quantization
problem16. Quarks have charges +-e/3 and +-2e/3.What combination of three quarks could yield (a)a proton, (b) a neutron? , (a) To obtain a charge of +1 with three quarks requires two charges of +2e/3 and one of charge –e/3 . Three quarks with charge + e/3 would violate the Pauli Exclusion Principle for spin 1/2 particles. (b)\ To obtain a charge of zero we could have either two with +e/3 and one with -2e/3 or one with+2e/3 and two with -e/3 . At this point in the text there is no reason to prefer either choice.
