From Newton to Einstein: A guided tour through space and time. with Carla Cederbaum

Transcription

1 From Newton to Einstein: A guided tour through space and time with Carla Cederbaum

2 Outline of our tour Sir Isaac Newton

3 Why are the planets orbiting the sun? Earth Satur n Sun

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7 Why are the planets orbiting the sun? inert heavy

8 Why are the planets orbiting the sun? inert heavy

9 Why are the planets orbiting the sun? inert heavy

10 Newton s new math - rate of change/derivative - vectors: velocity, acceleration, force

11 Newton s law of gravity m = mass of planet M = mass of sun G = gravitational constant = distance planet to sun

12 How do we measure mass? mass

13 Outline of our tour Siméon Denis Poisson Sir Isaac Newton Pierre Simon Laplace

14 Transform Newton s ideas into math!

15 Vector calculus Idea: generalize calculus to 3-dimensional space!

16 Newton s idea revisited U = Newtonian potential of sun G = gravitational ti constant t = mass density =mass/volume = differential operator

17 Where is? U = Newtonian potential of sun m = mass of planet = a differential operator

18 What is now mass M? M = mass of sun = normal vector to surface

19 What is now mass M? Apply mathematical theorems (by Gauß and dstokes)

20 Summary New math allows to - write Newton s ideas as differential equation - express mass as an integral (using mathematical theorems)

21 Bottom Line Use new math to model gravitation mathematically. gives better methods for predictions helps understand gravity better Newton s new physics inspired new math!

22 Outline of our tour Sir Isaac Newton Pierre Simon Laplace Siméon Denis Poisson Carl Friedrich Gauß Bernhard Riemann

23 How can we measure curvature?

24 How can we measure curvature?

25 Curvature is important for:

26 Differential Geometry - studies curves and surfaces - generalizes vector calculus l - allows rigorous definition of curvature (in terms of derivatives)

27 Curvature - Curves can be curved. - Surfaces can be curved. - 3-dimensional space can also be curved! - Can even think about higher dimensional (curved) space!!

28 Outline of our tour Sir Isaac Newton Pierre Simon Laplace Siméon Denis Poisson Carl Friedrich Gauß Bernhard Riemann Albert Einstein

29 Why are the planets orbiting the sun?

30 General Relativity

31 Math allows to make predictions like

32 Einstein s theory - is called general relativity - uses ideas from differential geometry like curvature - describes gravitational effects by a differential equation

33 General relativity Main equation in space-time : c = speed of light R, Ric: measure curvature g: measures distance/angles T: describes matter

34 Describes the world Einstein s theory is consistent with many measurements: - bending of light - gravitational red shift -

35 Applications - General Positioning System - satellites - space travel

36 General relativity in every day life:

37 General relativity in every day life: matter curves space-time

38 General relativity in every day life:

39 General relativity in every day life: curvature influences movement

40 General relativity in every day life:

41 Bottom Line Again: Use math to model gravitation. gives better methods for predictions helps to better understand gravity Gauß/Riemann s new math allows to predict new physics!

42 Outline of our tour Sir Isaac Newton Pierre Simon Laplace Siméon Denis Poisson Albert Einstein Carl Friedrich Gauß Bernhard Riemann Jürgen Ehlers today

43 Can we forget about Newton? Naive Idea: Yes! Einstein s general relativity is much better (in predicting observations)

44 Can we forget about Newton? Reconcile the theories:

45 Example: What is mass in general relativity? Negative mass? Many different definitions Hawking At infinity? ADM

46 What is a good local definition of relativistic mass? Step 1: differential geometry

47 Mass in general relativity new formula for mass (analogy to Newtonian formula): U,,, constructed t from geometry of space-time

48 Theorem [C. 11] Let and on every surface in a static space-time. Then.

49 Step 2: Newtonian limit Newton s theory: c=infinite Einstein s i theory: c= km/s 000k Newtonian limit: take c to infinity

50 Theorem [C. 11]

51 When is relativistic mass approximatively Newtonian mass? Result: If a star or black hole does not move then its relativistic mass is approximately equal to its Newtonian mass.

52 How do we find center of mass? Newton: center of mass

53 What is the center of mass in general relativity? Many different definitions Huisken- Huang Yau Metzger ADM All at infinity

54 What is a good local definition of relativistic center of mass? Step 1: differential geometry

55 CoM in general relativity new formula for center of mass (analogy to Newtonian formula): U,,,, constructed t from geometry of space-time

56 Theorem [C. 11] Let and on every space-time. surface in a static Then =

57 Theorem [C. 11]

58 When is relativistic center of mass approximatively Newtonian center? 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.

59 Step 1: get Bottom Line

60 What is the Newtonian Limit? See movie

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