Time happens that our consciousness moves There I object, said Filby. Of course a You must follow me carefully. I shall have to intermittently the latter from solid body may exist. All real things controvert one in or one two direction ideas thatalong are almost the beginning to the end our lives. So most people think. But wait a moment. universally accepted. The of geometry, for instance, That, saidyou a very youngisman, making spasmodic Can an instantaneous cube exist? they taught at school founded efforts to relight his cigar over the lamp; Don't follow you, said Filby. on a misconception. that . . very clear indeed. Is .not that rather a large thing to
Can a cube that does not last for any Now, is veryupon? remarkable that this expect us toitbegin said Filby, an is time at all, have a real existence? so extensivelyperson overlooked, the argumentative with redcontinued hair. Filby became pensive. Clearly, the Time Traveller, a slight accession of I do not meanwith to ask you to Time Traveller proceeded, any real body cheerfulness. is what is meant accept anythingReally withoutthis reasonable must have extension in four directions: it by the Fourth Dimension, ground for it. You will soonthough admit some as must have Length, Breadth, Thickness, people talkfrom about theYou Fourth much
aswho I need you. know of and - Duration. But through a natural 1895 Dimension not know they course that ado mathematical line,mean a lineit.ofIt is infirmity of the flesh, which I will explain only another wayno ofreal looking at Time. thickness nil, has existence. They taught to you in a moment, we incline to There is Neither no difference between time plane. These you that? has a mathematical overlook this fact. There are really four and any the abstractions. three dimensions of space things areof mere dimensions, three which we call the three planes except our consciousness moves Thatthat is all right, said the Psychologist.
of Space, and a fourth, Time. There is, however, a along it.having only length, breadth, and Nor, tendency to draw an unreal distinction between the thickness, can a cube have a real existence. former three dimensions and the latter, because it Time is the Fourth Dimension Time is the Fourth Dimension Our way of thinking about motion: Gods way of thinking about motion: You are three-dimensional and change in time You are four-dimensional; you have extension in time t y y x x Time and the Distance Formula 2 2 2 2 2 2 s x y z k tt Time is the fourth dimension Why isnt it in the distance formula? Have to get the units right What is k? k has units of velocity squared
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 2 s x y z k t 2 They both agree it is moving with velocity c x 2 2 y z t 2 2 2 x c 2 2 2
y z 2 c t 2 2 2 2 x y z c 2 t 0 x y z c 2 t k c 2 They agree on the value of s2 2 2 2 2 2 2 2 x y z k t x y z k t 2 2 2 2 s 2 x y z c 2 t 2 2
Solid Red Line Memorize this formula The 4D Distance Formula 2 2 2 2 2 s x y z c t 2 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 ct is the Fourth 2 Note that s can be positive or negative Dimension Cant always take the square root It is s2 that is invariant, so we simply dont take the square root. More on this later c = 2.99792458 108 m/s c = 3.00 108 m/ s Events and Space-Time Diagrams 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 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 P = (x,y,z,ct) ct x Proper Distance and Proper Time 2 2 2 2 2 s x y z c t 2 The quantity s 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 cant take the square root Instead, we define the proper time When s2 = 0, the separation is said to be lightlike. We dont bother taking the square root 2 2 2 2 2 2 2 2
c c t x y z s 2 P1 P2 Proper Distance and Proper Time 2 2 2 s 2 x y z c 2 t 2 2 2 2 2 2 2 c c t x y z 2 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 doesnt matter. P1 P2 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 Elsewhere (s2 > 0) Here and now Future Light Cone (s2 = 0) space Past Light Cone (s2 = 0) Elsewhere (s2 > 0) 2 2 2 2 2 s x y z c t 2 Absolute Past (s2 < 0) Good vs. Bad Coordinate Transforms 2 2 2 2 2 s x y z c t 2
If a coordinate transformation leaves the quantity s2 unchanged, then it must be good, and natures laws are the same in the original and final systems. Space Translation (x-direction) x' = x - a y' = y, z' = z, t' = t Time Translation x' = x , y' = y , z' = z t' = t - a Rotating Coordinates (around z-axis) x' = x cos + y sin y' = y cos - x sin z' = z , t' = t Galilean Boost (x-direction) x' = x - vt y' = y, z' = z, t' = Sample Problem 1.2 2 2 2 2 2 s x y z c t 2 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. 2 2 2 s2 x 0 y 0 z 0 c 2 t 0
2 2 Lorentz Boosts (x-direction) x' = x cosh - ct sinh y' = y z' = z ct' = ct cosh - x sinh 2 2 x cosh ct sinh y z ct cosh x sinh 2 x 2 cosh 2 2 xct cosh sinh c 2t 2 sinh 2 y 2 z 2 c 2t 2 cosh 2 2 xct cosh sinh x 2 sinh 2 x 2 cosh 2 sinh 2 c 2t 2 sinh 2 cosh 2 y 2 z 2 x 2 c 2t 2 y 2 z 2 s 2 What Does a Lorentz Boost Mean? ct Lorentz Boosts x' = x cosh - ct sinh y' = y , z' = z ct' =ct cosh - x sinh ct' x' x 0 x cosh ct sinh , x cosh ct sinh , x t c tanh , v c tanh Gods view: Its like a space-time rotation Primed observer is at x = 0 Our view: The primed oberver is
moving at velocity v = c tanh Rewriting the Lorentz Boost x x cosh ct sinh , ct ct cosh x sinh . sinh cosh tanh v c x x ct tanh ct 2 1 tanh 2 1 v c ct x tanh 2 1 tanh x ct v c 2 ct x v c 1 v 2 c 2 1 2
1 v c Note: 1 Dashed line: Know how to use it if given to you 2 tanh 2 1 tanh 1 2 1 tanh x x vt y y z z t t vx c 2 , , 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 wont necessarily be the same as measured by another observer (t'1 t'2) There is no agreement on ct simultaneity! Which of two events came first can be ambiguous! 1 1 v 2 c 2
x x vt y y z z ct' t t vx c 2 x' x Is Causality Lost? Although observers disagree on the order of events, they agree on the absolute future and the absolute past My worldline Absolute Future (s2 < 0) time Provided objects/ influences never travel faster than light, things can only Elsewhere be influenced by the absolute past, and (s2 > 0) can only influence the absolute future Here and now Elsewhere (s2 > 0) A world line is the path an object takes through space and time Future Light Cone (s2 = 0) space Past Light Cone (s2 = 0) Absolute Past (s2 < 0) Good Coordinate Transforms 2 2 2
2 2 s x y z c 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 2 Rotating Coordinates (around z-axis) x' = x cos + y sin y' = y cos - x sin z' = z , t' = t 1 1 v 2 c 2 Rotations (any axis), Lorentz boosts (any direction), Translations (space or time), and combinations of these Lorentz Boost (x-direction) x' = (x vt) y' = y, z' = z, t' = (t-vx/c2)
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