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

time spac e Spacetime diagram in the car’s rest frame

time spac e “car’s wordline” Spacetime diagram in our rest frame

time spac e “light cone”

time spac e Star Space Place, Monday, 7:15PM Ice Cream 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

time spac e 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