"IB Mathematics HLInternal AssessmentIs Time Travel possible? Table of ContentsAbstract ........................................................................................................................................... 3Introduction ..................................................................................................................................... 3Theory .............................................................................................................................................. 3Time dilation and proper time ........................................................................................................ 4The derivative: ............................................................................................................................. 6The limit: ...................................................................................................................................... 7The graph: .................................................................................................................................... 7For example - ............................................................................................................................... 8Solution: ...................................................................................................................................... 8Length & Mass dilatation .............................................................................................................. 10Length Dilation .............................................................................................................................. 11Conclusion: .................................................................................................................................... 12Citation: ......................................................................................................................................... 13 AbstractTime travelling has been a topic of discussion for a long time. Duringdifferent attempts to understand the “time travel” concept there wereidentified several paradoxes that could appear. Several time travel methodshave been identified and are based on pre existent physical laws andconcepts but none of them have been tried out or proved to be possible. Timetravel seems impossible but in the far future it could be achieved.IntroductionThroughout the ages “time travelling” has proved to be a topic raisingdifferent divergent opinions. During our life we all make mistakes and wishwe could go back in time to fix them or gointhefuturetoseewhat’s comingforus.But is time travel possible? We all travel in time. Duringthe last year, I’ve moved forward one year and so have you. Another way tosay that is that we travel in time at the rate of 1 hour per hour. But thequestion is, can we travel in time faster or slower than “1 hour per hour “?Orcan we actually travel backward in time, going back say 2 hours per hour or10 or 100 hours per hour?Theory One of the four dimensions is time. Time is usually perceived as a definitionto the period that occupies a life or space itself. The truth is time is notperceived the same by everyone it differs depending on the speed of theperson through space. Another part of the theory of relativity is thedimensional fabric called space-time when anything that has mass sits onthis fabric it causes a curvature path which is known as gravity so this alignswith the fact that gravity can bend time. In the theory of relativity, timedilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from agravitational mass or masses. In a sense this means that astronauts are timetravelers because every time they come back they would have traveled intime with a very small value. The 4 main ways to travel through time arewormholes, black holes, infinite cylinder, and cosmic strings. Wormholesbetween points in space and time follow Einstein’s equation but are onlysuitable for small particles and they collapse quickly. Another thing to noteis that creating wormholes is far beyond our technological capabilities.Black Holes is also another possibility of time travel since they causedistortions in gravity which can affect time. If a person hangs around theblack hole then time would move slower for them thus making them travelforward in time. Infinite Cylinder was proposed by astronomer Frank Tiplerproposed a mechanism where one would take matter that is 10 times thesun's mass, and then rolls it into very long but very dense cylinder. Afterspinning this up a few billion revolutions per minute, a spaceship nearbyfollowing a very precise spiral around this cylinder could get itself on a"closed, time-like curve", according to the Anderson Institute. CosmicStrings are narrow tubes of energy stretched across the entire length of theever-expanding universe. They are said to contain huge amounts of mass andtherefore could warp the space-time around them.Cosmic strings are either infinite or they’re in loops; they have no ends. Twosuch strings parallel to each other would bend space-time so dynamicallyand in such a specific manner that might make time travel possible.Time dilation and proper timeWhen someone moves very fast, time will flow slower - time is "diluted".The reason for this can be shown using light bouncing back and forward between mirrors on a moving train (a hypothetical device, sometimes calleda 'light clock'). For Example Two mirrors are a distance d apart and for Y on the same train it takes thetime? t’ for light to move between the mirrors (from one to the other).Ydd = C?TV?TThis time is the "proper" time = the time between two events measured inthe reference frame where they occur at the same place. Possibility For X not on the train which moves at v relatives to X the light have thesame speed (this is what relativity is all about!) But since the mirrors aremoving sideways it has a longer distance to travel. Since, ? ? ? ? ? ? ? ?? ? ? ? ?= ,? ? ? ? The only possibility left is that the time difference for X. The time it takesfor the light signal to move from one mirror to the other is ?t' as measuredby X; we can then find a triangle where the hypotenuse is the distancetravelled by the light according to X, that is c?t one other side is the distance travelled by X = V x ?t we also have the distance between the mirrors whichas measured by Y = c?t'.Pythagoras rule the gives(c ?t)² = (v ?t)²+(c?t')² c² ?t² = v² ?t² + c² ?t'² Dividing the above equation by c² gives,? ² ?t'² = ?t² - [] ?t² ? ² Factorizing out ?t² on the right side gives ? ² ?t'² = ?t² (1 - )? ² Taking square root on both sides, 1 ? ²2 ?t'= ?t(1 - ) ? ² Solving for?t1 ? ²2 ?t = ?t' (1 - )? ² The above equation can be written as ?t= ? ?t where? is Lorentz factor.0 The derivative:' ' ? ? ? -? '? Formula for derivative using quotient rule= ? ? ² 2 -( 1-? ) ' ' ? ? =2 ( 1-? )² ' ' ?? = Substitute 2 ? ? 2 1-? ' ? ? = 2 ( 1-? )² ? '? ? =2 2 (1-? ) 1-? ? ?0,1;? ² ?0,121-? = 12 1-? = 0? ' ? ? = = 02 2 (1-? ) 1-? The limit:X-> 0 => v-> 0? ? ? ? = ?? ? ??(? -? ) X-> 1 =>v-> C? ? ? ? = + 8? ? ??(? -? ) The graph: For example - Radio SignalA spaceship is sent from earth with the velocity 0.5c. A ratio signal is sentafter it with the speed c relative to earth. What is the speed of the radiosignal relative to the moving ship?We choose the positive direction so that the radio signal travels in it fromearth to the space.Solution:We know the radio signal’s speed relative to earth here “at rest”.So, U x = c The ship speed relative to earth is V = 0.5 c and Ux'=? (? ? – ? )U ' =x 1- U x Vc ² (? – 0.5? ) =1- c x 0.5 ?c ² 0.5? = 1-0.5 0.5? =0.5 = c3210C0 . 5 CExample: A missile is shot at the speed 0.3c from earth and hits a spaceship at 0.6ccreative to the ship. How fast is the ship moving and in what direction ? Solution: Again we take the positive direction from the earth to ship and now have:Ux =0.3 c Ux' = 0.6 c V = ?We need to solve Ux'Ux- V Ux V =for v, first we multiply with 1-giving Ux V c² 1-c² Ux V Ux' (1- )= Ux- V c² Ux V ' ' Ux -Ux = Ux- V 2 c Ux V V - Ux' = Ux-Ux' c² Ux-Ux'V =U x × V 1- Ux'c²0.3c- 0.6c V = 0.6c × 0.3c1- c² -0.3 cV = 1- 0.18V = -0.37c Where the negative value indicates that the strip moves in the ship – earthdirection, towards the earth and has a head on collision with the missile, butat the relative speed 0.6c, not 0.3c + 0.37c as its classically would havebeen.Length & Mass dilatation The length of moving objects also shrinks in the direction in which theymove. Get to the speed of LIGHT (v-> C) (not really possible, but imagine ifyou could for a moment) and the object's length would shrink to zero.A clock in a moving frame will be seen to be running slow, or dilatedaccording to the Lorentz transformation. The time measured in the frame inwhich the clock is at rest is called the “proper time”.’ ’ If the time travel T = t -t is measured in the moving reference frame, the0 2 1 ’ ’ T = t -t can be calculated using the Lorentz transformation.0 2 1"