General Relativity

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General Relativity Curved Space

• by 
• Robert Nemiroff
•  Michigan Tech

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Physics X About This Course

• Officially "Extraordinary Concepts in Physics"
• Being taught for credit at Michigan Tech
• Light on math, heavy on concepts
• Anyone anywhere is welcome
• No textbook required
• Wikipedia, web links, and lectures only
• Find all the lectures with Google at
• "Starship Asterisk" then "Physics X" 
• http//bb.nightskylive.net/asterisk/viewforum.php?
  f39

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General RelativityOverview

• GR Gravity is not a force
• It is a non-Euclidean geometry
• Very close to Newtonian gravity
• Tests
• precession of Mercury
• deflection of starlight
• Mechanism Actually -- none!  
• "...there is no theory of gravity beyond the
  mathematical form."  - Richard Feynman

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General Relativity History

• Created by Einstein between 1905 and 1915.
• Published incorrect theory of gravity in 1912.
• Test of incorrect theory clouded out.
• Published GR in 1915.
• Key GR test passed in 1919.
• Remains our most accurate theory of gravity.
• Many scientists feel that SR would have
  developed without Einstein, although Einstein got
  there first, but GR would not have been developed
  without Einstein. 

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General Relativity  Principles and Theorems

• "Local" effects are different than "global"
  effects
• Local very nearby 
• Space near mass "curves"
• Time near mass "slows"
• Equivalence Principle
• All things fall together
• Birkhoff's Theorem
• Ignore symmetric outside stuff
• Gravitomagnetism
• is to G as B is to E.

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General Relativity Concept of Spacetime

• Spacetime
• Space and time combined into one coordinate
  system
• time like a fourth spatial dimension 
• still three spatial dimensions
• lines represent coordinate system

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General Relativity  Principles and Theorems
You are in a small elevator.  Is it possible to
tell if you are accelerating downward without
gravity or near a gravitating mass?   1.  No. 2.
 Yes. 3.  Only if the elevator eventually
stops.
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General Relativity  Principles and Theorems
1.  No.   In GR, pure linear acceleration is
indistinguishable from gravity.   This is a
statement of the equivalence principle.
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General Relativity  Principles and Theorems
You are in a small elevator.  Can you tell the
difference between being near a gravitating mass
or in a space station spinning to produce
"artificial gravity"? 1.  No. 2.  Yes. 3.  Only
if "small elevator" is a euphemism for "physics
hell".
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General Relativity  Principles and Theorems
  2.  Yes.   The artificial "gravity" created by
rotation is inherently different than gravity,
although it may feel the same at first.  One way
to tell the difference is to rapidly spin some
object.  The object will try to maintain its
original spin axis.  Therefore, if the object
precesses, you might be on a spinning space
station.
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General Relativity Equivalence Principle
A hammer and feather both fall at the same
speed. Youtube video of Astronaut showing this
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General Relativity Equivalence Principle

•  
• "It is only when there is numerical equality
  between the inertial and gravitational mass that
  the acceleration is independent of the nature of
  the body." - Einstein
• Weak Equivalence Principle
• All objects fall the same in a vacuum.

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General Relativity Equivalence Principle

•  
• Strong Equivalence Principle
• ALL experiments appear the same when done 
• in a gravitational field, or
• in a linearly accelerating laboratory.
• G should be the same in the early universe.

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General Relativity Intervals

• Static, symmetric spacetime
• ds2 g11 dr2 g22 d?2 g33 dF2 g44 dt2
•  
• ds2 0 is a "null path" where photons fly.

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General Relativity Spacetime Intervals

• Two events happen in spacetime.  They are
  separated in space by ?r and in time by ?t.
•  
•  Time-like interval
• One event can affect the other
• c ?t gt ?r 
• Space-like interval
• One event cannot affect the other
• c ?t lt ?r
• Light-like interval
• Light from one could affect the other
• c ?t ?r
•   

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General Relativity Spacetime Motion

• Two events happen in spacetime.  They are
  separated in space by ?r and in time by ?t.
•  
•  Closed timelike curve (?r 0 ?t lt 0)
• A curve that comes back to the same spatial point
  at an earlier time (or later time)
• backwards time travel allowed in GR
• need other physics to disallow?
• will cover time travel in other lectures 
•   

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General Relativity   Energy Conditions

• Energy is conserved locally in GR.
• Consequence of time symmetry
• Consequence of Noether's theorem
• Energy is NOT conserved globally in GR.
• Example cosmological redshift -- where does the
  energy go?
• Potential energy is really just a bookkeeping
  device.  
• Not defined generally in GR.

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General Relativity The Key Equation

• Principle Equation
• Guv is the Einstein tensor 
• shows how spacetime curves
• guv is the metric tensor 
• shows spacetime geometery
• Tuv is the Stress Energy tensor
• shows energy and energy flow
• G, c, ? are all constants

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General Relativity Curved Space

• Flat Space
• familiar Euclidean geometry holds
• circles have area p r2
• spheres have volume (4/3) p r3
• Space curved by energy in GR
• Euclidean geometry does NOT hold
• Open space
• Area gt p r2 Volume gt (4/3) p r3
• Closed space
• Area lt p r2 Volume lt (4/3) p r3
•  

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General Relativity Curved Space
Example Triangles   Top Triangle area is less
than Euclidean    Middle Triangle area is more
than Euclidean   Bottom "Flat."  Triangle area
is exactly Euclidean.