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

16 views8 pages

Published on 10 Feb 2014

School

Department

Course

Professor

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 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 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

1

•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

2

•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 star’s 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

3