Time

1. “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 the Time Traveller, with a slight accession of cheerfulness. “Really this is what is meant by the Fourth Dimension, though some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between time and any of the three dimensions of space except that our consciousness moves along it.” H.G. Wells The Time Machine 1895 “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 founded on a misconception.” “Is not that rather a large thing to expect us to begin upon?” said Filby, an argumentative person with red hair. “I do not mean to ask you to accept anything without reasonable ground for it. You will soon admit as much as I need from you. You know of course that a mathematical line, a line of thickness 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. Time “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 explain to 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

2. Time is the 4th dimension Our way of thinking about motion: You are three-dimensional and change in time y x t God’s way of thinking about motion: You are four-dimensional; you have extension in time y x

3. Time and the distance formula The speed of light is always c, independent of the motion of the source or of the observer • 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

4. Two observers watching a light beam • They both agree it is moving with velocity c • They agree on the value of s2 Solid Red Line – Memorize this formula

5. The 4D distance formula • 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 ct is the Fourth Dimension c = 2.99792458  108 m/s c = 3.00 108 m/s

6. Events • A 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) What might the blue line represent? A) An event B) A small object at rest C) A small object moving at finite speed D) A small object moving at infinite speed Space-time diagrams ct • 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

7. Proper distance and proper time • 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

8. Proper distance and proper time • 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.

9. Past and Future in Relativity From 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? Absolute Future (s2 < 0) time Future Light Cone (s2 = 0) Elsewhere (s2 > 0) space Here and now Past Light Cone (s2 = 0) Elsewhere (s2 > 0) Absolute Past (s2 < 0)

10. Good vs. Bad Coordinate Transforms 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 sin y’ = y cos - x sin z’ = z , t’ = t Space Translation(x-direction) x’ = x - a y’ = y, z’ = z, t’ = t Time Translation x’ = x , y’ = y , z’ = z t’ = t - a Galilean Boost (x-direction) x’ = x - vt y’ = y, z’ = z, t’ = t

11. Lorentz Boosts Lorentz Boosts (x-direction) x’ = x cosh - ct sinh  y’ = y z’ = z ct’=ct cosh - x sinh  Prove the following The 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.

12. What does a Lorentz Boost mean? Lorentz Boosts x’ = x cosh - ct sinh  y’ = y , z’ = z ct’=ct cosh - x sinh  ct’ ct x’ God’s view: It’s like a space-time rotation  Primed observer is at x’ = 0 x Our view: The primed oberver is moving at velocity v = c tanh

13. Rewriting the Lorentz Boost Note:   1

14. Consequences of Lorentz Boost • 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! ct’ ct x’ x

15. Is Causality Lost? My worldline • 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 Absolute Future (s2 < 0) time Future Light Cone (s2 = 0) Elsewhere (s2 > 0) Here and now space Elsewhere (s2 > 0) Past Light Cone (s2 = 0) A world line is the path an object takes through space and time Absolute Past (s2 < 0)

16. Good Coordinate Transforms Rotating Coordinates(around z-axis) x’ = x cos + y sin y’ = y cos - x sin z’ = z , t’ = t Space Translation(x-direction) x’ = x - a y’ = y, z’ = z, t’ = t Time Translation x’ = x , y’ = y , z’ = z t’ = 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