I gave this presentation in 2019 at the Bear Creek School in Redmond to a group of very precocious high school students.

1. Einstein’s Road To Relativity
by Prashanth Adhikari
prashanth.adhikari@gmail.com

2. The Global Positioning System
https://www.slideshare.net/dr.geophysics/map-projections-datums-gis-and-gps-for-everyone

3. GPS Based Geolocation
https://www.slideshare.net/dr.geophysics/map-projections-datums-gis-and-gps-for-everyone

4. Relativistic Effects on GPS
• Clocks slow down when traveling fast.
• Clocks slow down in the presence of
a strong gravitational field.
• As a result, GPS clocks tick faster than
earth bound clocks by 38
microseconds per day.
• Therefore, clocks are offset to run
slower (10229999.9954326 Hz vs
10.23 MHz).
• Without relativistic corrections,
navigational errors would be > 11
km/day !
Relativity is important to our daily lives!
http://mathscinotes.com/2015/06/compensating-gps-clocks-for-the-effects-of-relativity/

5. Relativity goes to Hollywood

7. Physics circa 1900
• Universal laws of motion.
• Principle of inertia.
• F = ma
• Action and Reaction.
• Universal law of Gravitation
Isaac Newton James Clerk Maxwell
• Electromagnetic theory.
• Light as an electromagnetic wave.
Speed of light
Inconsistent!

8. Frames of Reference
Galilean Transformation
Lorentz Transformation
Michelson-Morley Experiment
(Only an approximation)

9. Special Relativity (1905)
Postulates:
• Principle of Relativity: The laws of physics are the same in all
inertial reference frames.
• Constancy of Speed of Light: The speed of light is a constant in
all inertial reference frames .
Consequences:
Time Dilation Length Contraction Relativistic Mass

10. Minkowski Space-Time
Euclidean Metric
Lorentz Metric
Newtonian ”Absolute”
Spacetime
• Lorentz Transformations
preserve the Lorentz Metric.
• No signal can travel faster
than the speed of light.
https://simple.wikipedia.org/wiki/Minkowski_spacetime#/media/File:World_line.svg
Lightlike region
Timelike Region
Spacelike Region

11. Limitations of Special Relativity
• Special status given to Inertial Frames of
Reference (but what about non-inertial).
• Newtonian gravity is not invariant under
Lorentz transformations.
• Newtonian gravity involves “instantaneous
action at a distance”, which violates special
relativity.
It took Einstein 10 years to solve this puzzle!

12. Foundations of
the General
Theory of
Relativity

13. Special Relativity and its
Generalization.

14. The Principle of General Covariance
The differential equations
describing the laws of
nature must be tensor
differential equations that
are invariant under
coordinate transformations.Mach’s Principle

15. The Principle of Equivalence
“The happiest thought of my life!”
- c 1907
Inertial mass = Gravitational mass

16. The need for Non-Euclidean Geometry
K
K’

17. Gauss’ Theory of Surfaces
Gauss Curvature
First Fundamental Form (Metric)
Second Fundamental Form
Theorema Egregium: Curvature is an intrinsic
property (depends only on the metric properties
of the surface and not on the embedding).

18. Curvature using planes

19. Examples of Surfaces
K < 0
K = 0
K > 0

20. Curvature and Topology
Euler Charactersistic:
Gauss-Bonnet Theorem:

21. Gauss-Bonnet for interior angles

22. Geodesics and Normal Coordinates
• Geodesics are analogues of straight lines.
• Often they are curves of shortest distance
(e.g. segments of great circles on a sphere).
• They can be used to choose a local coordinate
system where metric becomes flat at a point
(E =1, F = 0, G =1).
• This is analogous to choosing a local inertial
frame of reference.

23. Curved Spacetime
Spacetime tells matter how to move, matter tells spacetime how to curve. – John Wheeler

24. Riemannian Geometry
https://en.wikipedia.org/wiki/Riemannian_geometry#/media/File:G
eorg_Friedrich_Bernhard_Riemann.jpeg
• Vast generalization of Gauss’ theory to n
dimensions.
• Studies intrinsic geometry of “manifolds”.
• Riemannian metric (“line element”):
g_ij is a matrix of n X n functions on a local
coordinate system that encapsulates the geometry
of a manifold (for example, geodesics and
curvature).
Bernhard Riemann
First fundamental form for surfaces
Example: http://mathworld.wolfram.com/Hypersphere.html

25. Einstein’s Field
Equations of
Gravitation

26. Early Confirmation
Perihelion Precession of Mercury
Gravitational Lensing
Gravitational Redshift

27. A postcard from Schwarzschild

29. LIGO Interferometer

30. First Ever Image of a Black Hole

31. Thank you!

32. Appendix

33. Harnack House, Berlin

34. Max Planck
Einstein with Franck Werner Heisenberg

35. The Einstein Exhibition

36. The Manuscripts

37. The Manuscripts In
Space

38. The Einstein Centenary

39. Inside the Harnack House

40. Physics and Ideology

41. Einstein Conference
Yvonne Choquet-Bruhat Roger Penrose David Gross