“You must follow me carefully. I shall have to controvert one or two ideas that are almost universally accepted. The geometry, for instance, they taught you at school is foundedon a misconception.” “Is not that rather a large thing toexpect us to begin upon?” said Filby, an argumentative person with red hair. “I do not mean to ask you toaccept anything without reasonableground for it. You will soon admit asmuch as I need from you. You know ofcourse that a mathematical line, a line ofthickness nil, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.” “That is all right,” said the Psychologist. “Nor, having only length, breadth, and thickness, can a cube have a real existence.” “There I object,” said Filby. “Of course a solid body may exist. All real things –”

“So most people think. But wait a moment. Can an instantaneous cube exist?” “Don't follow you,” said Filby.

“That,” said a very young man, making spasmodic efforts to relight his cigar over the lamp; “that . . . very clear indeed.” “Now, it is very remarkable that this is so extensively overlooked,” continued theTime Traveller, with a slight accession of cheerfulness. “Really this is what is meantby the Fourth Dimension, though somepeople who talk about the FourthDimension do not know they mean it. It isonly another way of looking at Time.There is no difference between timeand any of the three dimensions of spaceexcept that our consciousness movesalong it.”

H.G. WellsThe Time Machine

1895

“Can a cube that does not last for any time at all, have a real existence?” Filby became pensive. “Clearly,” the Time Traveller proceeded, “any real body must have extension in four directions: it must have Length, Breadth, Thickness, and - Duration. But through a natural infirmity of the flesh, which I will explainto you in a moment, we incline to overlook this fact. There are really four

dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.”

Time is the Fourth Dimension

Time is the 4th dimension

Our way of thinking about motion:

God’s way of thinking about motion:

x

yYou are three-dimensional and change in time

x

y

t

You are four-dimensional; you have extension in time

Time and the distance formula 2 2 22s x y z

Time is the fourth dimension•Why isn’t it in the distance formula?•Have to get the units right•What is k?•k has units of velocity squared

2t 2

k t The speed of light is always c, independent

of the motion of the source or of the

observer

Two observers watching a light beam 2 2 2 22s x y z k t

•They both agree it is moving with velocity c

2 2 2x y z

ct

2 2 2x y z

ct

2 2 2 22 0x y z c t 2 2 2 22x y z c t

•They agree on the value of s2

2 2 2 2 2 2 2 2x y z k t x y z k t

2k c

2 2 2 22 2s x y z c t Solid Red Line – Memorize this

formula

The 4D distance formula 2 2 2 22 2s x y z c t

What does it mean?•c is a conversion constant, like inches/meter or seconds/day•The minus sign is really significant – it makes things fundamentally different•Note that s2 can be positive or negative

•Can’t always take the square root•It is s2 that is invariant, so we simply don’t take the square root.•More on this later

c = 2.99792458 108 m/s

c = 3.00 108 m/s

ct is the Fourth Dimension

EventsA point in space and time is called an event•It has both space and time coordinates

•We multiply time by c so everything can have the same units

P = (x,y,z,ct)

Space-time diagrams•Real graphs of the universe should be four-dimensional•We will draw two-dimensional graphs, one of space and one of time

•Time (ct) on vertical axis•Space (x) on horizontal axis

x

ct

What might the blue line represent?A) An eventB) A small object at restC) A small object moving at finite speedD) A small object moving at infinite speed

Proper distance and proper time

2 2 2 22 2s x y z c t

•The quantity s2 will be agreed on by all observers•s2 can be positive, negative, or zero•When s2 > 0, the separation is said to be spacelike

•The quantity s is called the proper distance•When s2 < 0, the separation is said to be timelike

•We can’t take the square root•Instead, we define the proper time

•When s2 = 0, the separation is said to be lightlike.•We don’t bother taking the square root

1P

2P

2 2 2 22 2 2 2c c t x y z s

Proper distance and proper time

2 2 2 22 2s x y z c t

•When s2 > 0, then there will always be an observer who sees the two events simultaneously with physical separation s•When s2 < 0, then there will always be an observer who sees the two events at the same place with time difference •Physicists debate which of these is the “correct” formula for distance, but in the end, it doesn’t matter.

2 2 2 22 2 2c c t x y z 1P

2P

Past and Future in RelativityFrom the viewpoint of a given event (here and now), which points will have positive values of s2, which will have zero, and which will have negative?

space

time

2 2 2 22 2s x y z c t

Here and now

Future Light Cone (s2 = 0)

Past Light Cone (s2 = 0)

Absolute Future (s2 < 0)

Absolute Past (s2 < 0)

Elsewhere (s2 > 0)

Elsewhere (s2 > 0)

Good vs. Bad Coordinate Transforms 2 2 2 22 2s x y z c t

If a coordinate transformation leaves the quantity s2 unchanged, then it must be good, and nature’s laws

are the same in the original and final systems.

Rotating Coordinates(around z-axis)

x’ = x cos + y siny’ = y cos - x sin

z’ = z , t’ = t

Space Translation(x-direction)

x’ = x - ay’ = y, z’ = z, t’ = t

Time Translationx’ = x , y’ = y , z’ = z

t’ = t - a

Galilean Boost(x-direction)

x’ = x - vty’ = y, z’ = z, t’ =

t

Lorentz Boosts 2 2 2 22 2s x y z c t

Lorentz Boosts(x-direction)

x’ = x cosh - ct sinh y’ = yz’ = z

ct’=ct cosh - x sinh

Prove the followingThe quantity s2 between an arbitrary event P1 = (x,y,z,ct) and the origin does not change when you perform a Lorentz boost in the x-direction.

2 2 2 22 20 0 0 0s x y z c t

2 22 2cosh sinh cosh sinhx ct y z ct x 2 2 2 2 2 2 2

2 2 2 2 2

cosh 2 cosh sinh sinh

cosh 2 cosh sinh sinh

x xct c t y z

c t xct x

2 2 2 2 2 2 2 2 2cosh sinh sinh coshx c t y z 2 2 2 2 2 2x c t y z s

What does a Lorentz Boost mean?Lorentz Boosts

x’ = x cosh - ct sinh y’ = y , z’ = z

ct’=ct cosh - x sinh

x

ct

x’

ct’

God’s view: It’s like a space-time rotation

Primed observer is at x’ = 0

0 cosh sinh ,x ct cosh sinh ,x ct

tanh ,x t c

tanhv c

Our view: The primed oberver is moving at velocity v = c tanh

Rewriting the Lorentz Boostcosh sinh ,

cosh sinh .

x x ct

ct ct x

2

2

tanhsinh ,

1 tanh

1cosh ,

1 tanh

2

2

tanh

1 tanh

tanh

1 tanh

x ctx

ct xct

22

22

1

1

x ct v c

v c

ct x v c

v c

tanh v c

2

x x vt

y y

z z

t t vx c

2 2

1

1 v c

Note:

1

Consequences of Lorentz Boost

2

x x vt

y y

z z

t t vx c

2 2

1

1 v c

•No observer can ever go faster than the speed of light in vacuum c!•The time t’ as measured by a moving observer depends on the position of the object being measured!•If two times are equal (t1 = t2) in one frame, they won’tnecessarily be the same as measured by another observer (t’1 t’2)

•There is no agreement onsimultaneity!

•Which of two events camefirst can be ambiguous!

x

ct

x’

ct’

Is Causality Lost?•Although observers disagree on the order of events, they agree on the absolute future and the absolute past•Provided objects/ influences never travel faster than light, things can only be influenced by the absolute past, and can only influence the absolute future

A world line is the path an object takes through space and time

space

time

Here and now

Future Light Cone (s2 = 0)

Past Light Cone (s2 = 0)

Absolute Future (s2 < 0)

Absolute Past (s2 < 0)

Elsewhere (s2 > 0)

Elsewhere (s2 > 0)

My worldline

Good Coordinate Transforms 2 2 2 22 2s x y z c t

Rotating Coordinates(around z-axis)

x’ = x cos + y siny’ = y cos - x sin

z’ = z , t’ = t

Space Translation(x-direction)

x’ = x - ay’ = y, z’ = z, t’ = tTime Translation

x’ = x , y’ = y , z’ = zt’ = t - a

Lorentz Boost(x-direction)

x’ = (x – vt)y’ = y, z’ = z,t’ = (t-vx/c2)

Rotations (any axis), Lorentz boosts (any direction), Translations (space or time), and combinations of these

2 2

1

1 v c