The Mathematics Behind Einstein’s Theory of Relativity Arick Shao The Wonderful World of Maths, Taster Day 4 April, 2017 Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 1 / 32
Introduction Who Is He? Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 2 / 32
Introduction Who Is He? Albert Einstein, physicist, 1879-1955 1905: Discovered special relativity. 1915: Discovered general relativity. Awarded Nobel prize in 1921. (1905: Discovery of the photoelectric effect.) Einstein in 1947 Photo by O. J. Turner. From the U.S. Library of Congress Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 2 / 32
Introduction Why Am I Here? Theory of relativity: Revolutionised modern physics. Involved advanced maths. Goal: Introduce maths behind: Special relativity 1 General relativity 2 Image of black hole from the movie Interstellar (Paramount). Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 3 / 32
Special Relativity Postulates and Definitions Einstein’s Postulates (A. Einstein, 1905) Postulates of special relativity: ∗ The laws of physics are the same in all inertial frames of reference. 1 The speed of light in vacuum has the same value in all inertial frames of reference. 2 Postulates ⇒ strange physical consequences Two observers moving at different velocities will: Perceive different events to be “at the same time”. 1 Measure different lengths for the same object. 2 Q. What is the mathematical explanation? Image from clipartall.com ∗ Quoted from Nobelprize.org. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 4 / 32
Special Relativity Postulates and Definitions Minkowski’s Contribution (Hermann Minkowski, 1907): Mathematical formulation of special relativity. In terms of geometry. Minkowski died soon after, in 1909. But his geometric viewpoint eventually led to ... ... Einstein’s theory of general relativity. H. Minkowski (1864–1909) Photo from www.spacetimesociety.org . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 5 / 32
Special Relativity Postulates and Definitions Spacetime t Classical physics—separate notions of: (3-d) space R 3 : contains points p = ( x , y , z ) . (1-d) time R : contains real numbers t . x , y , z Relativity—combined notion of spacetime. Notions of space and time cannot be separated. (4-d) spacetime R 4 : contains events P = ( t , x , y , z ) . Event P : “a point in space at a given time”. Observer: curve in spacetime (worldline). My worldline (yellow) in spacetime. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 6 / 32
Special Relativity Postulates and Definitions Classical Length and Distance Consider two points in space: p = ( p x , p y , p z ) , q = ( q x , q y , q z ) . q q − p p pq = q − p : vector from p to q . � | q − p | : distance from p to q . Squared distance from p to q : (Green) 3-vector from p to q . | q − p | 2 = ( q x − p x ) 2 + ( q y − p y ) 2 + ( q x − p x ) 2 . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 7 / 32
Special Relativity Postulates and Definitions Relativistic “Length” and “Distance” Consider two events in spacetime: q p = ( p t , p x , p y , p z ) , q = ( q t , q x , q y , q z ) . Now define “squared distance” from p to q by q − p m = −( q t − p t ) 2 + ( q x − p x ) 2 � q − p � 2 + ( q y − p y ) 2 + ( q x − p x ) 2 . p Similar to previous squared distance... ... but flip the sign in the t-component ! (Green) 4-vector from p to q . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 8 / 32
Special Relativity Postulates and Definitions And... the Dot Product Dot product in space: for 3-vectors � u = ( u x , u y , u z ) and � v = ( v x , v y , v z ) : � u · � v = u x v x + u y v y + u z v z . Captures lengths and angles. Generates (the usual) geometry on 3-dimensional Euclidean space. Spacetime product: for 4-vectors � u = ( u t , u x , u y , u z ) , � v = ( v t , v x , v y , v z ) : m ( � u , � v ) = − u t v t + u x v x + u y v y + u z v z . m ( � u , � u ) is the “weird squared length” of � u . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 9 / 32
Special Relativity Postulates and Definitions Minkowski Geometry Weird product ⇒ weird geometry on spacetime R 4 Called Minkowski geometry. Formally, described by the Minkowski metric: m = − dt 2 + dx 2 + dy 2 + dz 2 . (Here, we assumed units with speed of light c = 1.) Mathematically, we take this geometric viewpoint: Special relativity ⇔ Minkowski geometry. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 10 / 32
Special Relativity Postulates and Definitions Causal Character I m ( � u , � u ) can now be positive, negative, or zero! Each case has its own interpretation. (1) � u is spacelike: m ( � u , � u ) > 0 Points from origin to outside light cone. Represents spatial direction. Measures (squared) length/distance. (2) � u is null: m ( � u , � u ) = 0 Lies on light cone. Represents light ray. Image by Stib. From en.wikipedia.org . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 11 / 32
Special Relativity Postulates and Definitions Causal Character II (3) � u is timelike: m ( � u , � u ) < 0 Points from origin to inside light cone. Directions for observer worldlines. Measures (squared) time elapsed. Light rays are null lines. Bottom image: white ray. Observers are timelike curves. Bottom image: yellow curve. An observer can never travel faster than the speed of light! :( Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 12 / 32
Special Relativity Consequences Relativity Minkowski geometry is weird. Leads to interesting physical consequences. Many things cannot be measured absolutely: Examples: elapsed time, length, energy-momentum. Only makes sense relative to an observer. Image source unknown. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 13 / 32
Special Relativity Consequences Inertial Frames A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 Consider inertial observers A , B . Moving with constant velocities. Frame of reference ( t , x , y , z ) about A : x = y = z = 0 along A . A stationary. B moving relative to A . A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 Frame of reference ( ¯ t , ¯ x , ¯ y , ¯ z ) about B : x = ¯ y = ¯ z = 0 along B . ¯ B stationary. A moving relative to B . Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 14 / 32
Special Relativity Consequences Simultaneity I A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 Observers moving at different velocities see different events as “at the same time.” A 0 t = C B 0 Consider event A 0 on A ’s worldline. What A sees as simultaneous to A 0 is t = C ... ...i.e., space ( m -)perpendicular to A at A 0 . To A , events A 0 and B 0 are simultaneous. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 15 / 32
Special Relativity Consequences Simultaneity II A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 t = ¯ ¯ C A 0 But B sees differently! t = C B 0 t = ¯ What B sees as simultaneous to B 0 is ¯ C ... ...i.e., space ( m -)perpendicular to B at B 0 . But remember: this geometry is weird! A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 m -product is quite different. B 0 t = ¯ ¯ C does not hit A 0 . A 0 ¯ t = ¯ C t = C To B, events A 0 and B 0 not simultaneous! Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 16 / 32
Special Relativity Consequences Length Contraction I Observers moving at different velocities perceive lengths differently. Shaded region represents rod. A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 t = ¯ A stationary with respect to rod. ¯ C A 0 B moving with respect to rod. t = C B 0 Both A and B measure length of rod. A : length of green bolded segment. B : length of red bolded segment. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 17 / 32
Special Relativity Consequences Length Contraction II Note: A and B measure different lengths! Q. Who measures the longer length? (Hint: It’s a trick question.) A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 t = ¯ ¯ C A 0 t = C B 0 Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 18 / 32
Special Relativity Consequences Length Contraction II Note: A and B measure different lengths! Q. Who measures the longer length? (Hint: It’s a trick question.) B ’s measurement consists of: A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 A ’s measurement... t = ¯ ¯ C A 0 ... + timelike component, ... t = C B 0 ... which has opposite sign! B measures shorter length than A. Image from clipartkid.com Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 18 / 32
Special Relativity Consequences Time Dilation Clocks moving at different velocities observed to tick at different speeds. Both A and B carry a clock. A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 Clocks synchronised at O . A 0 A measures both clocks at t = C . t = c B 0 Q. What will A see? O A measures less time elapsed on B’s clock than A’s clock. Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 19 / 32
Special Relativity Consequences Twin Paradox I Twin paradox: classic thought experiment from special relativity. Consider twins, A and B . A goes to sleep for a long time. B flies off in a rocketship. B eventually flies back to A . Q. Who ages more? A or B ? Arick Shao (QMUL) Maths Behind Relativity Maths Taster Day 20 / 32
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