W.Murray PPD 1 Bill Murray RAL, STFC RAL 5 th November 2007 Maths: from a physicist Maths: from a physicist
W.Murray PPD 2 My brief Take a complex looking equation Make it look simple But which?
W.Murray PPD 3 E=mc 2 Einstein's famous equation Energy is related to mass The form is pretty simple What it means is fascinating physics How to derive it is a bit complex Lets use another equation Still not going to be easy
W.Murray PPD 4 Time Dilation Einstein's special relativity has Time Dilation Time slows down as you approach the speed of light
W.Murray PPD 5 Is it hard? The PHYSICS is crazy - Time and space mixed up Speed of light speed limit The maths looks a bit messy But it is only Pythagoras' Theorem
W.Murray PPD 6 Pythagoras Theorem a b c
W.Murray PPD 7 Proove Pythagoras? Think of a triangle a b c
W.Murray PPD 8 Proove Pythagoras? Take 4 of them
W.Murray PPD 9 Proove Pythagoras? Make a big square like this: a a a b b b b+a b2b2 a2a2
W.Murray PPD 10 Proove Pythagoras? Then slide them like this: a b+a b c c c2c2
W.Murray PPD 11 QED a b c c c2c2 a a b b b b2b2 a2a2 =
W.Murray PPD 12 Back to Einstein's Relativity 3 basic ideas: A) Things keep going in a straight line unless pushed Newton would agree B) The laws of physics are the same wherever you are Newton would probably agree C) The speed of light is a constant This was an experimental result
W.Murray PPD 13 Consider a light on a train A bulb flashes on one side And the light is detected on the other bulb Distance d me
W.Murray PPD 14 How long did it take? Think of me on a train I make a light flash on one side of the carriage And detect it at the other side Bristol to Paddington can get boring! How much later? I know how wide the train is Use speed = distance/time Speed of light is called 'c' The time is the width of the carriage over the speed of light
W.Murray PPD 15 You at Didcot station From Platform 1 things look different: Speed bulb
W.Murray PPD 16 You at Didcot station From Platform 1 things look different: A little later the train has moved along Speed bulb
W.Murray PPD 17 How long does it take? Well....how far does the light go? bulb d vt
W.Murray PPD 18 How long does it take? Time is distance over speed of light t=l/c, but we already said t=d/c How can both be true? d vt l
W.Murray PPD 19 How to make sense? We don't have time to discuss, But... On the train and off the train are describing the same thing Light going from bulb to detector And the say it takes different times So time is going at different rates!! By an amount to do with Pythagoras theorem for the triangle
W.Murray PPD 20 How long does it take? Distance from Pythagoras Theorem L will be bigger than d Not by much for Great Western d vt l
W.Murray PPD 21 Lets write down the maths Me on the train: You on the platform: Next take d=c t me from top line: Messy? Yes. t you is on both sides of equation
W.Murray PPD 22 And simplify it From last slide
W.Murray PPD 23 And simplify it Divide by c 2
W.Murray PPD 24 And simplify it Take off brackets
W.Murray PPD 25 And simplify it Move v 2/ c 2 bit around to get the t you together
W.Murray PPD 26 And simplify it Collect t you bits
W.Murray PPD 27 And simplify it take square root
W.Murray PPD 28 And simplify it Divide and swap sides
W.Murray PPD 29 Confused? Well..never mind Remember the PHYSICS: Both see the same things Both think light goes at a constant speed One thinks it has to go further So they must disagree about how long it takes bulb d vt d l
W.Murray PPD 30 Conclusion So thats special relativity. All the rest is left as an exercise: mass increasing, length decreasing and E=mc 2 And its all done with Pythagoras theorem