By: Katie Thorne, Ben Gookin & Bob Niffenegger

Outline  The Beginning Newton relativity Newton relativity Galileo Galileo Special Relativity Special Relativity Train paradox Train paradox Gamma factor Gamma factor Lorentz Lorentz Invariance Invariance Transformations Transformations Maxwell’s Invariance Maxwell’s Invariance Einstein’s Famous equation Einstein’s Famous equation The Math The Math

Newtonian Relativity  Space and time were absolute  Light propagate through aether Regulate speed like air for planes Regulate speed like air for planes Light travels at different speeds Light travels at different speeds  1881 Albert Michelson tried to measure this Used “Michelson Interferometery” Used “Michelson Interferometery” Found no variance Found no variance

Galilean Invariance  All fundamental laws of physics are the same in all inertial frames of reference  Applied to mechanics, we get Galilean transformations

What is special relativity? Einstein’s laws of physics in the absence of gravity. It describes how objects move through space and time

This brings up an interesting concept…. Time is not a universal quantity which exists on its own, separate from space. This means that time is not the same in all reference frames.

Reference point: Train moving at speed of light Reference point: Platform that is stationary Mirror Light source

This gives us the equation for time dilation The gamma factor appears in other relativistic expressions

An example:

Lorentz Invariance  All non-gravitational laws must give same predictions when given: Two different reference frames Two different reference frames Moving relative to each other Moving relative to each other  All fundamental equations of physics must be Lorentz invariant

Lorentz Transformations  Speed of light the same in all reference frames  Transform space-time coordinates (x,y,z,t) in one reference frame A, to another A’ moving at velocity V relative to A

Maxwell’s Equations  When Lorentz transformations are applied to Maxwell’s equations, the remain the same.  Thereby showing that they are invariant  This in essential for General Relativity Speed of light is the same in all reference frames Speed of light is the same in all reference frames

Where does Einstein’s famous equation come into play? Newtonian definitions of momentum, energy, and mass are not conserved in Special Relativity We can make small modifications to account for relativistic velocities

"Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move." -John Wheeler

Quick Math Overview  Tensor Vector (X) which under transformation (T) obeys this rule Vector (X) which under transformation (T) obeys this rule Metric Tensor Metric Tensor Geodesic Geodesic Curved Geometry Curved Geometry (Riemann Geometry)(Riemann Geometry) Energy Momentum Tensor Energy Momentum Tensor

Einstein’s Equation “These equations appeared so complicated that when first formulated them in 1915, he did not believe that a solution would ever be found. He was therefore quite surprised when, only a year later, Karl Schwarzschild created the Schwarzschild solution.

Bibliography  “A Serious but Not to Ponderous Book About Relativity”. Sheider, Walter. Cavendish Press. Ann Arbor, MI  Lecture Notes from Intro to Gravitation, Alexander B. Kostinski, Michigan Technological University  Lecture Notes from Honors Physics III, Bryan H. Suits, Michigan Technological University  Scienceworld.Wolfram.com  “Lorentz Covariance”. Wikipedia  “Lorentz Transformations”. Wikipedia  “Galilean Transformations”. Wikipedia  “Special Relativity”. Wikipedia