From Newton to Einstein:
A guided tour through
space and time
with Carla Cederbaum
Outline of our tour
Sir Isaac
Newton
1643-1727
Why are the planets orbiting the sun?
Earth
Satur
n
Sun
Why are the planets orbiting the sun?
inert
heavy
Why are the planets orbiting the sun?
inert
heavy
hea
Why are the planets orbiting the sun?
inert
heavy
Newton‘s new math
- rate of change/derivative
- vectors:
velocity,
acceleration, force
velocity acceleration
Newton‘s law of gravity
m = mass of planet
M = mass off sun
G = gravitational constant
= distance planet to sun
How do we measure mass?
mass
Outline of our tour
Siméon Denis
Poisson 1781-1840
Sir Isaac
Pierre Simon
Newton
1643 1 2 Laplace
1643-1727
1749-1827
Transform Newton
Newton‘s
s ideas into math!
Vector calculus
Idea: generalize calculus to
3-dimensional
3 dimensional space!
Newton‘s idea revisited
U = Newtonian potential of sun
G = gravitational
it ti
l constant
t t
= mass density =mass/volume
= “differential operator”
Where is ?
U = Newtonian potential of sun
m = mass off planet
l
t
= a differential operator
What is now mass M?
M = mass of sun
= normal vector to surface
What is now mass M?
Apply mathematical theorems
(by
Stokes)
(b Gauß
G ß and
d St
k )
Summary
New math allows to
- write Newton
Newton‘s
s ideas as
“differential equation”
- express mass as an integral
(using mathematical theorems)
Bottom Line
• Use new math to “model”
model gravitation
mathematically.
– gives better methods for predictions
– helps understand gravity better
• Newton’s new physics inspired new math!
Outline of our tour
Siméon Denis
Poisson 1781-1840
Carl Friedrich
Gauß
1777 1855
1777-1855
Sir Isaac
Pierre Simon
Newton
1643 1 2 Laplace
1643-1727
1749-1827
Bernhard Riemann
1826 1866
1826-1866
How can we measure curvature?
How can we measure curvature?
Curvature is important for:
Differential Geometry
- studies curves and surfaces
- generalizes
vector
li
t calculus
l l
- allows rigorous definition of
curvature
(in terms of derivatives)
Curvature
- Curves can be curved.
- Surfaces can be curved
curved.
- 3-dimensional space can
also be curved!
- Can even think about higher
dimensional (curved) space!!
Outline of our tour
Siméon Denis
Poisson 1781-1840
Carl Friedrich
Gauß
1777 1855
1777-1855
Sir Isaac
Pierre Simon
Newton
1643 1 2 Laplace
1643-1727
1749-1827
Albert Einstein
1879-1955
Bernhard Riemann
1826 1866
1826-1866
Why are the planets orbiting the sun?
General Relativity
Math allows to make predictions like
Einstein‘s theory
- is called “general relativity”
- uses ideas
differential
id
ffrom diff
ti l
geometry like curvature
- describes gravitational effects
by a differential equation
General relativity
Main equation in “space-time”:
c = speed of light
R, Ric: measure curvature
g:
measures distance/angles
T:
describes matter
Describes the world
Einstein’s theory is consistent
with many measurements:
- bending of light
- gravitational red shift
- …
Applications
- General Positioning System
- satellites
t llit
- space travel
General relativity in every day life:
General relativity in every day life:
matter
tt curves space-time
ti
General relativity in every day life:
General relativity in every day life:
curvature
t
influences
i fl
movementt
General relativity in every day life:
Bottom Line
• Again: Use math to model gravitation
gravitation.
– gives better methods for predictions
– helps to better understand gravity
• Gauß/Riemann’s new math allows to
predict new physics!
Outline of our tour
Siméon Denis
Poisson 1781-1840
Carl Friedrich
Gauß
1777 1855
1777-1855
Sir Isaac
Pierre Simon
Newton
1643 1 2 Laplace
1643-1727
1749-1827
Bernhard Riemann
1826 1866
1826-1866
Jürgen Ehl
Jü
Ehlers
1929-2008
Albert Einstein
1879-1955
today
Can we forget about Newton?
Naive Idea: Yes!
Einstein‘s general relativity
i much
is
hb
better
tt
(in predicting observations)
Can we forget about Newton?
Reconcile the theories:
Example:
What is mass in general relativity?
Many different
definitions
Negative mass?
Hawking
At infinity?
ADM
What is a good local definition of
relativistic mass?
Step 1: differential geometry
Mass in general relativity
new formula for mass
(analogy to Newtonian formula):
U,
, , constructed
U
t t d from
f
geometry of
space-time
Theorem [C. ‘11]
Let
and
on every
surface in a static space-time.
Then
.
Step 2: Newtonian limit
Newton’s theory: c=infinite
Einstein’s
theory: c=300.000km/s
Ei t i ’ th
300 000k /
Newtonian limit:
take c to infinity
Theorem [C. ‘11]
When is relativistic mass
approximatively
i ti l Newtonian
N t i mass?
?
Result: If a star or black hole
does not move
then its relativistic mass
is approximately equal to
its Newtonian mass.
How do we find center of mass?
Newton:
center of mass
What is the center of mass
in general relativity?
Many different
definitions
HuiskenHuisken
Huang
Yau
Metzger ADM
All at infinity
What is a good local definition of
relativistic center of mass?
Step 1: differential geometry
CoM in general relativity
new formula for center of mass
(analogy to Newtonian formula):
U,
, , , constructed
U
t t d from
f
geometry of
space-time
Theorem [C. ‘11]
Let
and
on every
surface in a static
space-time.
Then
=
Theorem [C. ‘11]
When is relativistic center of mass
approximatively
i ti l Newtonian
N t i center?
t ?
Result: If a star or black hole
does not move
then its relativistic center of mass
is approximately equal to
its Newtonian center of mass.
Bottom Line
Step 1: get
What is the Newtonian Limit?
See movie