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Astronomy & Astrophysics
Michael Reid

S2.1 Einstein’s revolution – Special Relativity • An object is travelling RELATIVE TO sth • No object can travel faster than the speed of light • If you observe sth moving by you at a speed close to the speed of light, you’ll conclude time runs slowly for it. o A person moving by you ages more slowly than you, a clock moving by you ticks more slowly than yours; o the person in the spaceship moving by you thinks that you’re moving by/away from him at a speed close to the speed of light, your clock ticks more slowly o Both viewpoints are the same • The size of an object moving by you at a speed close to the speed of light, you’ll find its length is shorter/smaller than it’d be when it’s stationary • The mass of an object moving by you at a speed close to the speed of light, you’ll find its mass to be greater than the mass it’d have if it’s stationary • Fig S2.2b – up & down are relative to the centre of the Earth • 2 people share the same reference frame when they’re not moving relative to each other Thought experiments 1-9 - SEE slides for length contraction & time dilation Thought experiment 10 - Fig. S2.14 • Jackie is moving relative to you & her twin sister, you’ll see Jackie’s time running more slowly than yours & her sister’s. When you give a push to both Jackie & her sister, Jackie feels the push for a shorter time > Jackie’s momentum becomes smaller > smaller effect on Jackie’s velocity than her sister’s > from Newton’s 2 law (F = ma), smaller effect on velocity = smaller acceleration, given the same amount of force, mass of Jackie is greater S2.4 How does special relativity offer us a ticket to the stars? • Fig S2.19 – if the distance between Earth & Vega = 25 light-years, as you travel at 0.999c, relative to the people on the Earth, the trip to Vega takes just over 25 years; • From your point of view on the spaceship, you remain stationary while Earth rushes away from you at 0.999c & Vega approaches you at 0.999c > length between Earth & Vega contracted, say to about 1 light-year • Therefore, the round trip takes only 2 years to you but people on Earth will be 50 years older S3.1 Einstein’s 2 revolution - General relativity • Matter shapes the ‘fabric’of spacetime in a matter analogous to the way heavy weights distort a rubber sheet. The greater the mass, the stronger the distortion of spacetime • Tenets of general relativity o Gravity arises from distortions of spacetime o The stronger the gravity, the more slowly the time runs. o Black holes can exist in spacetime, & falling into a black hole means leaving the observable universe o The universe has no boundaries & no center, yet it might have a finite volume o Large masses that undergo rapid changes in motion or structure emit gravitational waves that travel at the speed of light 1 • The Equivalence Principle o The effects of gravity are exactly equivalent to the effects of acceleration  When we accelerate upward at g (9.8 ms-2) through space, we can feel weight as we do on Earth, thereby keeps you stationary; Jackie is weightless because she is in free-fall through the gravitational field (no upward force/acceleration to her) S3.2 What’s curved spacetime? • Actual geometry of spacetime = a mixture of flat geometry, spherical geometry & saddle-shaped geometry • According to the equivalence principle, we can attribute a feeling of weight either to experiencing a force generated by acceleration, or to being in a gravitational field. Similarly, when we weightless, we may attribute it either to being in free-fall, or to travelling at a constant velocity far from any gravitational fields o Since travelling at constant velocity = travelling in a straight line, objects experiencing weightlessness must be travelling in the straightest possible path between 2 points in spacetime o Our orbital speed keeps us going around instead of hitting the Sun. According to the equivalence principle, all orbits must represent the straightest possible path through spacetime. Shapes & speeds of orbits reveal the geometry of spacetime. S3.3 A new view of Gravity • Einstein’s general theory of relativity: Earth feels NO force tugging on it in its orbit, & therefore follows the straightest possible path through spacetime. o What we perceive as gravity arises from the curvature of time • Rubber sheet analogy o No masses: each pair of circles separated by an equal radial distance o Presence of a heavy mass (Sun): it forms a bowl-like depression; & circles become widely separated near the bottom of the bowl o Gravity becomes stronger & the curvature becomes greater as we approach the Sun’s surface o Marbles on this sheet follow the straightest possible path given the curvature of the sheet  Those moving slowly & close to the center would follow circular/elliptical orbits around the center, while those rolled from farther away/at higher speeds could loop around the center on unbound parabolic/hyperbolic paths o Explanations from these orbits are different from Newton’s law of gravitation – a mass like the Sun causes the spacetime to curve, & the curvature of spacetime determines the paths of freely moving planets • Weightlessness = following the straightest possible path, as long as engines are off & spacecraft are unaffected by atmospheric drag • Limitations of analogy o Rubber sheet is supposed to represent the universe, but masses should within the rubber sheet o Planets don’t orbit in precisely the same plane o Miss out time dimension 2 • How does gravity affect time? o When the spaceship is accelerating, its speed is constantly increasing relative to the outside observer, both you & Jackie are changing reference frames constantly o In an accelerating spaceship, time must run faster at the front end & more slowly at the back end. Thus, the light from Jackie’s flashes at the back will take a little longer to reach you at the front. You conclude time is running slowly at the back end of the spaceship; o When the spaceship is at rest under gravitational field, time would run slowly at the back end, applying the equivalence principle [gravitational time dilation] S3.4 Testing General Relativity • Newton’s law of gravity said time is absolute & space is flat. In reality, time runs more slowly & general relativity takes distortion of spacetime into account • Gravitational lensing o Observe trajectories of light rays o The curvature of space near the Sun should cause the light beam passing closer to the Sun to curve more than the light beam from the other star → apparent angular separation appear smaller than their true separation → a shift in the star’s apparent position o Gravitational lensing occurs when a massive object curves spacetime, altering trajectories of light beams passing nearby > light beams end up converging at Earth, distorting the appearance of the star/galaxy & creating multiple images of it The Twin Paradox • Involving identical twins, one of whom makes a journey into space in a high-speed spaceship and returns home to find that the twin who remained on Earth has aged more. • Not a paradox when resolved with special relativity • Because the twins are not equivalent; the space twin experienced additional, asymmetrical acceleration when switching direction to return home. Time runs more slowly in an accelerating spaceship (at the speed of light). (3X108X1s = very long time) CH 18.3 Black Holes: Gravity’s Ultimate Victory • Making an object of a particular mass more compact increases its escape velocity • Michell & Laplace found that objects are so compact that their escape velocities > speed of light; it’s possible that an object’s gravity is so strong that not even light can escape → black hole • Event horizon = boundary between the inside of a black hole & the universe outside o A boundary around a black hole at which escape velocity = speed of light o The boundary = spherical because the velocity needed to escape a black hole’s gravity depends on the distance to its center, which is the same for every point on the event horizon • Einstein’s general theory of relativity: gravity arises from curvature of spacetime • 2-D representation of spacetime (=Rubber Sheet Analogy) o No masses: each pair of circles separated by an equal radial distance o Presence of a mass: it curves spacetime & circles become widely separated o Curvature becomes greater & greater > black hole = bottomless; center lies within the event horizon 3 • Radius of event horizon = Schwarzchild radius, depending only on the mass of a black hole • Properties of a black hole o Mass:A collapsing stellar core becomes a black hole when it shrinks to a size smaller than Schwarzchild radius, the core disappears within its own event horizon. The black hole still contains all its mass & exerts gravity associated with that mass o Electric charge: quickly attract oppositely charged particles, making it electrically neutral o Angular momentum: according to conservation of angular momentum, a black hole should faster and faster as it shrinks in size • General relativity: any massive rotating object should drag neighboring regions of spacetime around in circles > accelerate infalling objects in the direction of rotation, preventing objects from falling straight downward. This can change the shape of event horizon, as matter & energy moving along with the black hole’s rotation can resist falling into the black hole • Fig 18.14 - Tidal forces would be lethal near a black hole as it pulls more strongly on the feet than on the head, stretching him lengthwise & squeezing him from side to side. o However, supermassive black holes make the tidal forces much weaker & human may survive. • Formation of a black hole o A stellar core collapses beyond the neutron star li
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