By:By:

Katie Thorne,Katie Thorne,Ben Gookin & Bob NiffeneggerBen Gookin & Bob Niffenegger

OutlineOutline The BeginningThe Beginning

Newton relativityNewton relativity Galileo Galileo

• Special RelativitySpecial Relativity• Train paradoxTrain paradox• Gamma factorGamma factor

• LorentzLorentz• InvarianceInvariance• TransformationsTransformations

• Maxwell’s InvarianceMaxwell’s Invariance• Einstein’s Famous equationEinstein’s Famous equation• The MathThe Math

Newtonian RelativityNewtonian Relativity Space and time were Space and time were

absoluteabsolute Light propagate through Light propagate through

aetheraether Regulate speed like air for Regulate speed like air for

planesplanes Light travels at different Light travels at different

speedsspeeds 1881 Albert Michelson 1881 Albert Michelson

tried to measure thistried to measure this• Used “Michelson Used “Michelson

Interferometery”Interferometery”• Found no varianceFound no variance

Galilean InvarianceGalilean Invariance

All fundamental laws All fundamental laws of physics are the of physics are the same in all inertial same in all inertial frames of referenceframes of reference

Applied to mechanics, Applied to mechanics, we get Galilean we get Galilean transformationstransformations

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

2

2

1'c

vtt

21

1

An example:

Lorentz Invariance Lorentz Invariance

All non-gravitational laws must give same All non-gravitational laws must give same predictions when given:predictions when given: Two different reference framesTwo different reference frames Moving relative to each otherMoving relative to each other

All fundamental equations of physics must All fundamental equations of physics must be Lorentz invariantbe Lorentz invariant

Lorentz TransformationsLorentz Transformations

Speed of light the same in all reference Speed of light the same in all reference framesframes

Transform space-time coordinates (x,y,z,t) Transform space-time coordinates (x,y,z,t) in one reference frame A, to another A’ in one reference frame A, to another A’ moving at velocity V relative to Amoving at velocity V relative to A

Maxwell’s EquationsMaxwell’s Equations

When Lorentz transformations are applied When Lorentz transformations are applied to Maxwell’s equations, the remain the to Maxwell’s equations, the remain the same.same.

Thereby showing that they are invariant Thereby showing that they are invariant This in essential for General RelativityThis in essential for General Relativity

Speed of light is the same in all reference Speed of light is the same in all reference framesframes

Where does Einstein’s famous equation come into play?

E m c 2

•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

ji

jii xTx

2

1

Quick Math OverviewQuick Math Overview

TensorTensor Vector (X) which under Vector (X) which under

transformation (T) transformation (T) obeys this ruleobeys this rule

Metric TensorMetric Tensor GeodesicGeodesic Curved GeometryCurved Geometry

• (Riemann Geometry)(Riemann Geometry) Energy Momentum Energy Momentum

Tensor Tensor

),( jiij eeg

2

221

21

21

211

1''

21 xTxT

xTxTxx

2223

22

21

2 tcxxxs

jiij xxgs 2

Einstein’s EquationEinstein’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.

22

2222212

2 )2

1())((sin)2

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rc

MGs

BibliographyBibliography

““A Serious but Not to Ponderous Book A Serious but Not to Ponderous Book About About Relativity”. Sheider, Walter. Relativity”. Sheider, Walter. Cavendish Cavendish Press. Ann Arbor, MI. 1996Press. Ann Arbor, MI. 1996

Lecture Notes from Intro to Gravitation, Alexander B. Lecture Notes from Intro to Gravitation, Alexander B. Kostinski, Michigan Technological UniversityKostinski, Michigan Technological University

Lecture Notes from Honors Physics III , Bryan H. Lecture Notes from Honors Physics III , Bryan H. Suits, Michigan Technological UniversitySuits, Michigan Technological University

Scienceworld.Wolfram.comScienceworld.Wolfram.com ““Lorentz Covariance”. WikipediaLorentz Covariance”. Wikipedia ““Lorentz Transformations”. WikipediaLorentz Transformations”. Wikipedia ““Galilean Transformations”. WikipediaGalilean Transformations”. Wikipedia ““Special Relativity”. WikipediaSpecial Relativity”. Wikipedia