The Twin Paradox Tyler Stelzer Bob Coulson Berit Rollay A.J. Schmucker Scott McK

The Twin Paradox Tyler Stelzer Bob Coulson Berit Rollay A.J. Schmucker Scott McK

The Twin Paradox Tyler Stelzer Bob Coulson Berit Rollay A.J. Schmucker Scott McK

Garin, Nina, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal The Twin Paradox Tyler Stelzer Bob Coulson Berit Rollay A.J. Schmucker Scott McKinney “When you sit with a nice girl as long as two hours, it seems like two minutes. When you sit on a hot stove as long as two minutes, it seems like two hours, that’s relativity.“ -Albert Einstein The Twin Paradox “If I had my life to live over again, I’d be a plumber.“ -Albert Einstein Overview Events in addition to Cooridinatizations; The concept of Spacetime Lorentz Coordinatizations; Lorentz Postulates Minkowski Space LorentzTrans as long as mations Moving Reference Frames Time Dilation Length Contraction Lorentz-Einstien Trans as long as mations Boosts The Twins Paradox

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Events in addition to Coordinizations The Concept of Spacetime Events: What are they An “event” is a definite happening or occurrence at a definite place in addition to time. Examples: A bomb explodes Emission of a photon, (particle of light), by an atom; similar to switching a light on in addition to off. What is Spacetime Let E be the set of all events as previously defined. This set is call Spacetime. Let e represent a particular event, then “e E” means e is an event.

Modeling Spacetime To model spacetime we use R4. The idea behind this is that each event e E is assigned a coordinate (Te , Xe , Ye , Ze ) R4 R4 is an ordered four-tuple Example: (x,y,z,w) has 4 coordinates What Are Te , Xe , Ye , Ze Te – The time coordinate of the event. Xe – The x position coordinate of the event. Ye – The y position coordinate of the event. Ze – The z position coordinate of the event. What does this mean e (Te , Xe ,Ye , Ze ) is assumed to be a bijection. This means that: It is a one to one in addition to onto mapping Two different events need to occur at either different places or different times. Such assignment is called a “Coordinatization of Spacetime”. This can also be called a “Coordinatization of E”.

Lorentz Coordinizations & Postulates Vector position functions in addition to Worldlines In Newtonian physics/calculus, moving particles are described by functions t r(t) r(t) = ( x(t) , y(t) , z(t) ) This curve gives the “history” of the particle. Vector position functions in addition to Worldlines cont View this from R4 perspective t ( t, r(t) ) In the above ‘t’ represents time in addition to ‘r(t)’ represents the position. This can be thought of as a “curve in R4”, called the Worldline of the particle.

What is time There are 2 types of time Physical Clock Time Coordinate Time: the time furnished by the coordinatization model: e ( Te , Xe , Ye , Ze ) 1st Lorentz Postulate For stationary events, Physical Clock Time in addition to Coordinate Time should agree That is, we assume that stationary st in addition to ard clocks measure coordinate time. 2nd Lorentz Postulate The velocity of light called c = 1. Light always moves in straight lines with unit velocity in a vacuum. T (T, vT + r0), time in addition to spatial position Note: Think of the light pulse as a moving particle.

Minkowski Space Geometry of Spacetime Minkowski Space (Geometry of Spacetime) The symmetric, non-degenerate bilinear as long as m of the inner product has the properties = = + = c The inner product does not have to be positive definite, which means the product of it with itself could be negative. Non-degenerate meaning only the zero vector is orthogonal to all other vectors Spacetime has it’s own geometry described by the Minkowski Inner Product. Minkowski Inner Product Defined on R4: u = (u0,u1,u2,u3) v = (v0,v1,v2,v3) :=u0v0- u1v1- u2v2- u3v3 < , > also called the Lorentz Metric, the Minkowski metric, in addition to the Metric Tensor M = R4 with Minkowski Inner Product “ ” represents the usual inner product (dot product) in R3 In this case you have an inner product that allows negative length.

How is the Minkowski Inner Product Related to the Euclidean Inner Product The Euclidean Inner Product: r = (r1, r2, r3) s = (s1, s2, s3) r s =(r1 s1 + r2 s2 + r3 s3) Note: R4 = R1 x R3 The Minkowski Inner Product u = (u0,(u1,u2,u3)) v = (v0,(v1,v2,v3)) :=u0v0- (u1,u2,u3) (v1,v2,v3) Strange Things Can Happen In Minkowski Space Such as: Vectors can have “negative lengths” Non-Zero vectors can have zero length. A vector v M is called: “Time Like” if > 0 “Null” if = 0 (Some of these are Non-Zero Vectors with zero length.) “Space Like” if < 0 (These are the negative length vectors) Minkowski Space serves as a mathematical model of spacetime once a Lorentz coordinization is specified. Consider an idealized infinite pulse of light. Consider the Problem of Describing Light We think of a moving light pulse as a moving particle emitted via a flash in spacetime. The path of this particle is referred to as it’s worldline. By the second Lorentz Postulate, the worldline is given by: T (T, vT + r0) Recall: v v = 1 ( v R3) r0 R3 (T, vT + r0) = (T, vT) + (0 , r0) = T(1, v) + (0 , r0) Note: T(1, v) is a null vector because < (1, v) , (1, v)> = 1- v v = 1-1=0 (This is an example of a Non-Zero Vector with zero length.) a:=(1,v) b:=(0, r0) So, the worldline of a light pulse will be of the as long as m TaT+b M with = 0 (These are called null lines.)

Light Cones Suppose b M The light cone at b:={p M = 0} This is the union of all null lines passing through b. The as long as ward light cone at b = {P=(P0,P1,P2,P3) M P light cone at b, P0-b0>0} The backward light cone at b = {P=(P0,P1,P2,P3) M P light cone at b, P0-b0 <0} Moving Reference Frames Moving Reference Frames Recall the idea of a “coordinatization” e E, e (te, xe, ye, ze) “e” O C B A Z Y X Garin, Nina San Diego Union-Tribune Features Reporter

The Idea: By trig, determine “spatial coordinates” (xe, ye, ze) Assuming: c = speed of light, Rate X Time = Distance, Time at which the light pulse reaches O “e” Interesting Math Problem Suppose there is a 2nd coordinate system, moving at a constant velocity v, in the direction of the x-axis. Suppose O, O’ both employ the same procedure as long as coordinatizing E: (T, X, Y, Z) (Stationary Frame) (T’, X’, Y’, Z’) (Moving Frame) How are these two frames related First, let’s look at a picture: Assume constant velocity c = 1

Solution: Assume O, O’ have st in addition to ard clocks. e.g. Einstein – Langevin clock (light pendelum) A rigid rod (or tube) of length L duration of time between emission in addition to return 1 unit of time = Time Dilation How does O regard O’’s clock Think of O’’s clock as sitting in a moving vehicle (e.g. a train or spaceship) Spaceship moving at a velocity v O’’s perspective Let t’ = time of ½ pendulum L = ct’ Distance = (Rate)(Time) Now Consider O’s Perspective Light Source Mirror L Mirror has moved since the ship has moved vt ct Let t be the time of the ½ pendulum of O’’s clock as observed by O. Observe: (ct)2 = L2 + (vt)2

References Relativistic Electrodynamics in addition to Differential Geometry, Parrot, Springer-Verloy 1987.

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