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
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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.
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2 )2
1())((sin)2
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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
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