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Spacetime diagrams
•A useful way of making spacetime maps
•Similar to normal map (north-south & east-west)...
• ...but with one axis being time instead of a direction
North
West
tim
e
space
Spacetime diagram in the car’s rest
frame
tim
e
space
“car
’s w
ordl
ine”
Spacetime diagram in our rest frame
tim
e
space
“ligh
t con
e”“light cone”
tim
e
space
Star party @ Space Place, Monday,
7:15PM
Ice Cream Social @ Michael’s, Sunday,
7:15PM
Gravity
•Special relativity formulated in inertial frames
Must find an inertial frame for SR!
For electric force, just use a neutral particle to anchor frame to, that will be unaccelerated.
But gravity accelerates everything (remember Gallileo?)
•Special relativity does not hold under gravity
Stationary Elevator with gravity: Ball is accelerated down
Outside of an accelerated elevator: Ball at rest
Inside of accelerated elevator: ball accelerated down
=elevator with gravity!
General relativity•Einstein’s fundamental
insight: “Equivalence principle”
Gravity accelerates everything
⇒ Gravity must be a property of spacetime
Gravity and acceleration are indistinguishable (Galileo)
⇒ Formulate physics in terms of accelerated frames
Equivalence principle
Elevator at rest
Elevator in uniform motion
Inside the moving elevator
Accelerated elevator from outside
Inside the accelerated elevator
= In an elevator in a gravitational field
Light bending:
•Gravity bends light
•Recall: light travels on spacetime geodesics
⇒ In spacetime with gravity, geodesics are curved
•Geodesics are the straightest possible lines
⇒ Gravity curves spacetime
Spacetime curvature
tim
e
space
Spacetime curvature
In curved space• Parallel lines don’t stay
parallel
• Triangles don’t add up to 180°
• The straightest possible lines are “geodesics”
• The stronger the curvature, the stronger theses effects
In curved spacetime
•The actual length to a destination is changed (try this yourself!)
•The circumference of a circle is no longer 2πR (try this yourself!)
•Sometimes, more than one path is the shortest path (try this yourself!)
Is space curved?
What curves spacetime?
•Gravity curves spacetime
•We know that mass causes gravity
⇒ Mass curves spacetime
What curves spacetime?
•Einstein’s most fundamental equation relates the curvature to mass:
More mass, more curvature
More curvature closer to mass
•Einstein’s equivalent to Newton’s law of gravity
“Field equation”
Light bending (2):•Heavy objects curve spacetime
•Galaxy clusters are very heavy:
1000 trillion times more massive than the sun
• They should curve spacetime a lot
• Light should follow curved path around them