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2903 Space (and Time)

For everything there is a season,
And a time for every purpose under heaven:
A time to be born, and a time to die;
A time to plant, and a time to reap;
A time to kill, and a time to heal;
A time to break down, and a time to build up;
A time to weep, and a time to laugh;
A time to mourn, and a time to dance;
A time to throw away stones, and a time to gather stones together;
A time to embrace, and a time to refrain from embracing;
A time to seek, and a time to lose;
A time to keep, and a time to throw away;
A time to tear, and a time to sew;
A time to keep silence, and a time to speak;
A time to love, and a time to hate,
A time for war, and a time for peace.
Ecclesiastes 3:1-8

Relative Motion

Suppose a train is travelling at 5 m/s and a bandit is running towards the front at 2 m/s, relative to the train.
How fast is he moving relative to the ground?

How does this motion appear from the point of view of someone in the train?

How does this motion appear from the point of view of the bandit?
The proper name for "point of view" is "frame of reference": a frame of reference which is not accelerating is an "inertial frame"
This is Galilean Relativity: All inertial frames are equivalent
We can consider doing an experiment in two different frames:
1. Earth Frame: if we measure a distance(velocity) in this frame, we will call it x(v)
2. Train Frame: if we measure a distance(velocity) in this frame, we will call it x'(v')

e.g. just dropping a ball

In the train frame

In the earth frame

Results of any experiment can be described in any frame: no frame is preferred.

Put differently: you cannot do an experiment to decide if you are moving, since one man's motion is another man's station!.
Can transform the results of an experiment in any one frame to any other.

Velocity in earth frame = Velocity of train frame + Velocity in train frame
\color{red}{ v = v_0 + v'}

and can compare them both

Have gone through this in detail since it is wrong!

1. Laws of Physics are the same in all inertial frames,
2. Time is the same in all frames
[2] is a hidden assumption, that was never written down. The correct statement (Einstein) is
1. Laws of Physics are the same in all inertial frames,
2. The speed of light is the same in all frames
This means that (since speed = distance/time) distance and/or time must change when we go from one frame to another.

Suppose we fire a beam of light from the front of a train.

From the point of view of the earth we would expect
\color{red}{ c = v_0 + c'}
in fact
\color{red}{ c = c'}

Time Dilation

To find out how the time changes from one frame to another, consider bouncing a light off a mirror as the train goes past.
In the train frame:
In the earth frame, the light has to travel further, since the train has moved.

We can solve this

giving
\color{red}{ t' = t\sqrt {1 - \frac{{v_0 ^2 }}{{c^2 }}} }
so \color{red}{t' < t} i.e. moving clocks run slow

Twin Paradox

e.g. Suppose you are in an OC Transpo bus (v₀ = 10ms^-1):

how slow will your watch appear to run compared to your clock at home?
e.g. A muon is a subatomic particle which lives 2x10^-6 s.
How long will it appear to live if is travelling at .99c?
Note that (1-v₀²/c²)^1/2 ~ 1 for all cases we are familiar with (which is just as well!)

Since time is not the same in two frames, events which are simultaneous in one frame are not in another.

e.g in the earth frame

and the ttrain frame

The twin paradox: α-Centauri is 4 light-years distant from earth. Fred and Fred' are both 21. Fred' leaves for α-Centauri at .9 c.
How old is Fred when Fred' gets back? How old is Fred'?

Your reaction to all this should be:

"This is really stupid. What really happens?"
Answer: In physics you cannot ask
"What really happens?"
The best one can do is ask
"What can I measure?"
Reality is a dangerous concept

So can we measure it?

Hafele-Keating dexperiment done in 1980: atomic clock flown round the world (first-class!) and compared to time of atomic clock "at rest". Time lost by moving clocks ~ 190 ns
Vladimir: That passed the time
Estragon: It would have passed in any case.
Vladimir: Yes, but not so quickly.
Beckett: waiting for Godot.

Black Holes

A particle will escape from the earth if it has positive energy

At the earth's surface
\color{red}{ v = \sqrt {\frac{{2GM}}{R}} }
However we can interpret this differently: what radius would the earth have for a given escape velocity?
\color{red}{ R = \frac{{2GM}}{{v^2 }}}
In particular, if the escape velocity is the speed of light c, nothing can escape
\color{red}{ R = \frac{{2GM}}{{c^2 }}}
This is the Schwarzchild radius (loosely the black-hole radius) for any mass.
What is this for the earth?
Statutory Warning:This is a fudge: you cannot treat light as a massive particle, nor can you handle a very strong gravitational field as if it were a weak one......
(there are actually two factors of 2 error which cancel out.....weren't we lucky!)

Time and Again

Time as an extra dimension.

The changes to space and time that Einstein found show that they are aspects of the same thing: space-time.
Galileo-Newton Space is 3-D, time is an independent quantity
Einstein-Minkowski, Space-time is 4-D, and motion mixes space and time in different ways Einstein's concept of time can be expressed graphically by "worldlines" in a space-time diagram.

Reduced to 1 (2) space and 1 time dimension, can describe interactions as events: e.g 2 men walk into each and fall over.

Relativity tells us that the following constaints must be satisfied by world-lines:

Must be oriented from past to future "flow of time".
Static object remains at same x, but time still moves.
Moving objects cannot have a slope > c (otherwise object would be moving faster than light.)
Events occur when worldlines intersect

Possible light paths are rep. by "Light Cone". Cannot escape the light cone (it includes all possible futures for you!)

"Light Cone" represents possible paths of signals: i.e. interaction need not be direct, but can send (e.g.) phone message. Shows person A sending signal at time t₁ which is received by B at time t₂ Note times are measured in A's frame

Can see the violation of simultaneity: e.g. two flashes of light are seen as simultaneous by observer A but not by B

Some world-lines are more complex: e.g. a planet with 2 space dimensions

with 1 space dimension

Gravity modifies all this: e.g. close to a black hole, all your futures include falling into it!

Statutory Warning: We have represented time as a 4th dimension: this doe not mean it is the fourth dimension.

e.g suppose we have an event now and one in the future at time t and position x: the distance is not

\color{red}{ s^2 = t^2 + x^2 }

(in fact we can't even add space and time).

We can use

\color{red}{ s^2 = x^2 - c^2 t^2 }

(note the minus sign) but even this needs careful interpretation. e.g. Suppose we emit a flash of light

However this 4-D picture is useful. We can analyse the time in any creative work in the same way:

e.g. Midsummer Nights Dream.
Note this is a gross over-simplification: e.g the lovers + fairies + Bottom have very complex crossing world-lines
The assertion: prior to 1900 the space-time diagram for any work satisified the standard conditions: Aristotle's three unities become "space-time causality is preserved" or "special relativity is satisified".

The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line
, Nor all thy Tears wash out a Word of it.
Rubaiyat of Omar Khayyam tr. by Edward FitzGerald
Time past and time present
Are always present in time future.
T. S. Eliot Burnt Norton

And the seasons they go round and round
And the painted ponies go up and down
We are captive on the carousel of time
We cant return we can only look behind
From where we came
And go round and round and round
In the circle game
Joni Mitchell The Circle Game

(Ian and Sylvia)

After 1900, a variety of works violate this in a non-trivial way:
e.g. Three Time Plays by J. B Priestley: "Time and the Conways"
Could we have a story in which a murder is committed in one frame but not another?
Some suggestions:
- Dangerous Corner J. B Priestley
- I have Been Here Before J. B Priestley
- All You Zombies (Robert Heinlein)
- Back to the Future
- Einstein's Dreams
- Sliding Doors

Einstein's next question to himself was "what happens if we are in an accelerating frame?"

Why do all masses fall at same rate? All normal forces (e.g. electrical, friction, elastic...) don't produce same accn in all bodies.
\color{red}{ F = m_I a}
The inertial mass m_I) measures how hard things are to accelerate (2nd. law)
\color{red}{ F = m_G g}
the second (gravitational mass m_G) measures gravitational force
but we know everything falls at teh same rate (well, in a vacuum) so
\color{red}{ m_I = m_G = m \Rightarrow a = g}
Are we really sure the m's are the same? This concerned Newton
T the second (gravitational mass m_G) measures gravitational force
F = m_Ia F = m_Gg

Pseudo-forces (e.g. centrifugal force) behave the same way.
Maybe gravity is somehow a fictitious force (?!?!?!?)
```
m_I = m_G

F = m_Ia = m_Ig 
```
so a = g only if the "inertial mass" ≅ "gravitational mass".
Can demonstrate this is true to 1 part in 10¹² (Eötvos experiment).
Special relativity said you cannot do an experiment to decide if you are moving.
General says that you cannot do an experiment to distinguish between a gravitational field and an acceleration (!!!!!!!!!)

For example:

Suppose you are in a stationary elevator, and a bullet is shot horizontally, it will fall due to gravity..

Suppose you are in an accelerating elevator, and a bullet is shot horizontally, it will appear to fall..

Unless you look at it in the earth frame

You cannot distinguish them

For example:

Suppose you are in an accelerating elevator, and a beam of light is shot horizontally, it will appear to fall..
Suppose you are in a stationary elevator, and a beam of light is shot horizontally, it will fall..
You cannot distinguish the two. Light gets affected by gravity?

General relativity:

Handles frames which are accelerating w.r.t. each other.

What is a straight line?
Which is the straight line?

Dynamics:

Why bother?

A Body continues at rest or in a state of uniform motion unless acted on by a force.

Uniform motion means in a straight line.
A geodesic in Euclidean space ≡ straight line ≡ shortest path.

Can either say:
1. There is a force called gravity which acts on all energies (and hence attracts light)
2. There is no such thing as gravity, it's just that masses distort space-time in their neighbourhood
Massive bodies follow timelike geodesics so planets are actually moving in "straight" lines in a curved space...
Either way, don't jump off tall buildings: you can be just as dead in a curved space!
"Lenses extend unwish through curving wherewhon till unwish returns on its unself" e.e.cummings

Gravitational Red-shift

A ball thrown up near the earth's surface will lose energy.

Again can get this via equivalence principle:

light emitted from floor of elevator hits ceiling after a time t = h/c.
during this time, lab (elevator) has accelerated to a speed u = gt,
so the light gets stretched out
This is another consequence of the equivalence principle: confirmed in numerous experiments over the last 40 years. Implies that clocks run slow in gravitational fields $$ \color{red}{ t' = \frac{t}{{\left( {1 - \frac{{2GM}}{{c^2 r}}} \right)^{1/2} }}} $$
A consequence: time stops at the edge of a black-hole for an external observer.

And light gets does bent:

this is a very large cluster of galaxies, which acts as a very large (and rather bad!) lens. It produces several images of a much more distant galaxy

Note that bending of light by gravity allows two (or more) geodesics: can imagine that this allows two possible time-lines reaching same point

e.g. Dangerous Corner (Priestly) although there the play is set up so the time-lines are sequential
Sliding Doors (Peter Howitt)
will return to this idea later on in Quantum Mechanics

Geometry of Curved spaces

Note we have carefully avoided saying what we mean by a curved space

If you take the example of the 2-D curved surface of the Earth, this is embedded in a 3-D space. Hence If a massive body curves space, it almost implies extra dimensions.
In fact we can carry out tests to decide if we live in a "normal" 3-D space (Euclidean) e.g.
parallel lines
angles of a triangle add up to 180⁰
α + β + γ = π
These are experiments that we can almost do. (Gauss tried the 2nd!).

So can we build a time-machine?

Godel produced a model universe consistent with GR with closed time-lines (not like ours: it has a centre is not homogenous and rotates, ours has no centre,seems to be homgenoues and doesn't rotate.
Tipler showed that can construct time machine from infinite rotating massive cylinder
Light cone gets bent round cylinder, so starting point lies inside
But it requires negative energy: now known (see Ford and Roman Sci Am. article) that negative energy allows time-travel, so probably can't construct one.

If we could construct a time machine, can we evade the paradoxes?

Various possibilites:

Free will doesn't exist.
"Among the things Billy Pilgrim could not change were the past, the present, and the future."

"Out of 31 inhabited planets I have visited, only on Earth is there any talk of free will."

Vonnegut., Slaughterhouse Five
can we prove free-will?
The past is immutable: i.e. we could see it like a movie, but not an interactive game.

Does time "flow"

You could not step twice into the same river; for other waters are ever flowing on to you. Heraclitus

River analogy is not uncommon: e.g

Time is a sort of river of passing events, and strong is its current; no sooner is a thing brought to sight than it is swept by and another takes its place, and this too will be swept away. Marcus Aurelius

and books:

"Randall and the River of Time" (C. S. Forester)
"Riverworld" (Philip Jose Farmer)
and music
"From me flows that which you call time" (Takemitsu)
"Of time and rivers flowing
The seasons make a song"
Peter Seeger

But in reality the analogy of time flowing is not very useful: flow of soemthign is "rate of change" e.g. flow of a river is amount of water that passes youi in a given time. If time is to flow, it is the amount of time that passes you in a given time .... Seems to require "meta-time" If it doesn't flow, is it continuous?

the final issue; Why does time have a direction?

parallel lines
angles of a triangle add up to 180⁰ α + β + γ = π These are experiments that we can almost do. (Gauss tried the 2nd!).