AST201H1 Study Guide - Midterm Guide: Weightlessness, Gravitational Time Dilation, Electromagnetic Spectrum

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Published on 10 Feb 2014
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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.
oA person moving by you ages more slowly than you, a clock moving by you ticks more
slowly than yours;
othe 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
oBoth 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 sisters. 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
sisters > from Newton’s 2nd 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 2nd 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
oGravity arises from distortions of spacetime
oThe stronger the gravity, the more slowly the time runs.
oBlack holes can exist in spacetime, & falling into a black hole means leaving the observable
universe
oThe universe has no boundaries & no center, yet it might have a finite volume
oLarge masses that undergo rapid changes in motion or structure emit gravitational waves
that travel at the speed of light
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The Equivalence Principle
oThe 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
oSince 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
oOur 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.
oWhat we perceive as gravity arises from the curvature of time
Rubber sheet analogy
oNo masses: each pair of circles separated by an equal radial distance
oPresence of a heavy mass (Sun): it forms a bowl-like depression; & circles become widely
separated near the bottom of the bowl
oGravity becomes stronger & the curvature becomes greater as we approach the Sun’s
surface
oMarbles 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
oExplanations 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
oRubber sheet is supposed to represent the universe, but masses should within the rubber
sheet
oPlanets don’t orbit in precisely the same plane
oMiss out time dimension
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How does gravity affect time?
oWhen the spaceship is accelerating, its speed is constantly increasing relative to the outside
observer, both you & Jackie are changing reference frames constantly
oIn 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;
oWhen 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
oObserve trajectories of light rays
oThe 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 stars apparent position
oGravitational 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
oA boundary around a black hole at which escape velocity = speed of light
oThe 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)
oNo masses: each pair of circles separated by an equal radial distance
oPresence of a mass: it curves spacetime & circles become widely separated
oCurvature becomes greater & greater > black hole = bottomless; center lies within the event
horizon
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