2. May 8, 2001Lynn Cominsky - Cosmology A3501 Answer to last week’s Question Do you think there is a cosmic Creator? A) Yes, and the Creator was responsible for fine tuning the laws of physics to allow life B) Yes, and the Creator is still responsible for the entire Universe C) No, the physical laws just happened by accident D) This question can’t be answered with the available data D or ??

3. May 8, 2001Lynn Cominsky - Cosmology A3502 Group 13 Justin Beasley Tiffany Borders Erica Lillund

4. May 8, 2001Lynn Cominsky - Cosmology A3503 Hyperspace and Strings Lecture 13

5. May 8, 2001Lynn Cominsky - Cosmology A3504 Big Bang Timeline We are here Today’s lecture

6. May 8, 2001Lynn Cominsky - Cosmology A3505 Big Bang?

7. May 8, 2001Lynn Cominsky - Cosmology A3506 Distance in 3 Dimensions We all experience three spatial dimensions (usually referred to as x, y and z) Distances in three dimensions are easily found from an extension of the 2D Pythagorean theorem for right triangles a 2 + b 2 = c 2 In 3D: d = a 2 + b 2 + c 2

8. May 8, 2001Lynn Cominsky - Cosmology A3507 Vectors Vectors are used to mathematically represent quantities which have both size and direction This vector d has components (a, b, c) in the (x, y, z) directions and magnitude d d = abcabc

9. May 8, 2001Lynn Cominsky - Cosmology A3508 Vector Fields Vector fields are physical quantities which have magnitudes and directions at each point This is a 2D vector field where the direction at each point is given by What is the magnitude (length) of the vector at each point?

10. May 8, 2001Lynn Cominsky - Cosmology A3509 Tensors It is difficult to visualize a tensor This is a visualization of the stresses in a 3D material when force is applied at two points on the top surface The stress tensor is a 3 x 3 matrix of numbers Push here

11. May 8, 2001Lynn Cominsky - Cosmology A35010 Tensors Einstein unified the 3 spatial dimensions with the dimension of time to make a four dimensional space time (x, y, z, ct) in which gravity is defined by a 4x4 tensor Components of the tensor down the diagonal are the coefficients in d 2 = g 11 x 2 + g 22 y 2 + g 33 z 2 + g 44 c 2 t 2 They are (1,1,1,-1) for flat space

12. May 8, 2001Lynn Cominsky - Cosmology A35011 Tensor Fields The electromagnetic field is another example of a 4D tensor field. It has 4x4 components, which tell you the magnitude of the components in 3 different directions for both the electric and magnetic field. EM field 0 B z -B y -iE x -B z 0 B x -iE y B y -B x 0 -iE z iE x iE y iE z 0

13. May 8, 2001Lynn Cominsky - Cosmology A35012 Flatland It’s hard to visualize 4 dimensions, so let’s start out by examining the lives of the characters in Edwin A. Abbott’s Flatland Rank in Flatland is a function of increasing symmetry: A woman, soldier, workman,merchant, professional man, gentleman, nobleman, high priest

14. May 8, 2001Lynn Cominsky - Cosmology A35013 Flatland What do they see when a 3D being (Lord Sphere) comes to visit? 3D cross-sections of Lord Sphere float through the 2D world of Flatland

15. May 8, 2001Lynn Cominsky - Cosmology A35014 Troubles in Flatland It’s hard to eat in a 2D world! It is also impossible to tie your shoes! Why? A digestive tract cuts a 2D being in half!

16. May 8, 2001Lynn Cominsky - Cosmology A35015 Troubles in Flatland A Square and his wife alone in their 2D house, when Lord Sphere drops in from the third dimension There is no privacy in 2D from a 3D being!

17. May 8, 2001Lynn Cominsky - Cosmology A35016 Troubles in Flatland A 3D being would be able to change the symmetry of a 2D resident or help him escape from jail! The 3D being can lift the 2D resident up out of Flatland!

18. May 8, 2001Lynn Cominsky - Cosmology A35017 Troubles in Flatland How do Flatlanders know the shape of their Universe? A flat plane (with edges) is an open 2D Universe Is there a closed 2D Universe? A Moebius strip is a 2D closed universe movie

19. May 8, 2001Lynn Cominsky - Cosmology A35018 Troubles in Flatland What would happen if Flatlanders walked all the way around a closed 2D world? They would be mirror-reversed! Flat torusFlat torus – another example of a closed 2D world

20. May 8, 2001Lynn Cominsky - Cosmology A35019 The 3D Universe Open (  <1) Hyperbolic geometry Flat (  =1) Euclidean geometry Closed (  >1) Spherical geometry

21. May 8, 2001Lynn Cominsky - Cosmology A35020 Curved Space This is not an infinite series of reflections, but is caused by light traveling all the way around the hyperdonut A hyperdonut is one example of a curved space in 3D

22. May 8, 2001Lynn Cominsky - Cosmology A35021 The 4D Universe Many cosmologists believe that our Universe is a 4D hypersphere This is a 3D movie projection of a 4D hypersurface movie

23. May 8, 2001Lynn Cominsky - Cosmology A35022 Geometry in the 4 th dimension A 2D square is created by moving a line in a perpendicular direction A 3D cube is created by moving a square in a perpendicular direction

24. May 8, 2001Lynn Cominsky - Cosmology A35023 Geometry in the 4 th dimension A Flatlander can only visualize a cube, if it is unfolded in 2D If you move a 3D cube in a fourth perpendicular direction, you get a hypercube A 3D being can only visualize a hypercube by unfolding it in 3D into a tesseract

25. May 8, 2001Lynn Cominsky - Cosmology A35024 Geometry in the 4 th dimension Christus Hypercubus was painted by Salvador Dali in 1955 –it features a tesseract A 4D hypercube is bounded by 8 3D cubes, has 16 corners and a volume L 4

26. May 8, 2001Lynn Cominsky - Cosmology A35025 Geometry in the 4 th dimension Here is another 2D projection of a 4D hypercube At each face, you can see a cube in different directions as you change your perspective d 2 = x 2 + y 2 + z 2 + w 2

27. May 8, 2001Lynn Cominsky - Cosmology A35026 Troubles in Spaceland Thieves from the fourth dimension could steal things from locked safes (or operate without cutting you open!) There is no privacy in 3D from a 4D being!

28. May 8, 2001Lynn Cominsky - Cosmology A35027 Visitors from the 4 th dimension Try the “digustoscope” to see yourself as a 4D being in a 3D world! Do powerful beings such as a Cosmic Creator (or the Devil) live in the Fourth Dimension?

29. May 8, 2001Lynn Cominsky - Cosmology A35028 4D movies by Andy Burbanks Spinning hypercubes Inflated hypercube is almost a hypersphere Spinning hypertorus http://info.lboro.ac.uk/departments/ma/gallery/hyper/index.html 1 cube of 8 is white

30. May 8, 2001Lynn Cominsky - Cosmology A35029 Geometry in higher dimensions Here is 2D projection of a 5D hypercube It obeys the same mathematical laws as objects in worlds with a lower number of dimensions d 2 = x 2 + y 2 + z 2 + w 2 +v 2

31. May 8, 2001Lynn Cominsky - Cosmology A35030 Physics in higher dimensions Kaluza was the first to try to unify the fields of (Maxwell) electromagnetism and (Einstein) gravity by rewriting the laws of physics in 5D (or a 5x5 tensor) In this theory, light was caused by a ripple in the 5 th dimension

32. May 8, 2001Lynn Cominsky - Cosmology A35031 Kaluza-Klein Theory Theodor Kaluza’s original idea was refined by mathematician Oskar Klein At each point in 4D spacetime, another curled up or “compactified” dimension is present, but it is so small that it is not observable At each point in spacetime, a curled up dimension exists

33. May 8, 2001Lynn Cominsky - Cosmology A35032 Physics in Higher Dimensions Grand Unified Theory expands the 5x5 tensor to include the Yang-Mills field, which describes the weak and strong interactions in N dimensions The resultant tensor has 4+N dimensions N is 5 for the standard model of particle physics

34. May 8, 2001Lynn Cominsky - Cosmology A35033 Physics in Higher Dimensions Supersymmetry allows particles with different types of spin (fermions and bosons) to interchange Each particle has a supersymmetric partner called a “sparticle” No “sparticles” have yet been detected The WIMP (weakly interacting massive particle) is the lightest “sparticle”

35. May 8, 2001Lynn Cominsky - Cosmology A35034 Physics in Higher Dimensions Supergravity theory includes supersymmetry as well as interactions with matter (gravity) It requires an 11D Kaluza-Klein theory However, it still does not have enough complexity to explain all the interactions that we see in the Standard Model Meanwhile, searches for “sparticles” continue at higher energies

36. May 8, 2001Lynn Cominsky - Cosmology A35035 Superstrings Strings are little closed loops that are 10 20 times smaller than a proton Strings vibrate at different frequencies Each resonant vibration frequency creates a different particle Matter is composed of harmonies from vibrating strings – the Universe is a string symphony “String theory is twenty-first century physics that fell accidentally into the twentieth century” - Edward Witten

37. May 8, 2001Lynn Cominsky - Cosmology A35036 Superstrings Strings can execute many different motions through spacetime But, there are only certain sets of motions that are self-consistent Gravity is a natural consequence of a self- consistent string theory – it is not something that is added on later Self-consistent string theories only exist in 10 or 26 dimensions – enough mathematical space to create all the particles and interactions that we have observed

38. May 8, 2001Lynn Cominsky - Cosmology A35037 Superstring animations MSNBC web site Official string theory web site: http://superstringtheory.com/index.html http://superstringtheory.com/index.html Infinitesimal black hole turning into elementary particle via Brian Greene’s superstring theory

39. May 8, 2001Lynn Cominsky - Cosmology A35038 Superstring Dimensions Since we can observe only 3 spatial and 1 time dimensions, the extra 6 dimensions (in a 10D string theory) are curled up to a very small size The shape of the curled up dimensions is known mathematically as a Calbi-Yau space

40. May 8, 2001Lynn Cominsky - Cosmology A35039 Superstring Universe At each point in 3D space, the extra dimensions exist in unobservably small Calabi-Yau shapes

41. May 8, 2001Lynn Cominsky - Cosmology A35040 Superstring Interactions Strings interact by joining and splitting 2 stringsjoinedsplit into 2

42. May 8, 2001Lynn Cominsky - Cosmology A35041 Superstring Interactions Strings annihilate and erupt repeatedly subject to the quantum mechanical uncertainty principle, just like particle pairs 2 strings annihilation 2 virtual strings eruption 2 strings

43. May 8, 2001Lynn Cominsky - Cosmology A35042 Superstrings and Gravity Gravitational force is represented by the exchange of closed strings Even if an infinite number of string-like Feynman diagrams are added up, the theory stays finite

44. May 8, 2001Lynn Cominsky - Cosmology A35043 Superstring Motions Strings can have two different types of motions in a universe where some dimensions are curled up and others are extended Sliding Winding

45. May 8, 2001Lynn Cominsky - Cosmology A35044 Superstring Theories There are at least five different versions of string theory, which seem to have different properties As physicists begin to understand the mathematics, the different versions of the theories begin to resemble each other (“duality”) In 1995, Edward Witten showed how all five versions were really different mathematical representations of the same underlying theory This new theory is known as M-theory (for Mother or Membrane)

46. May 8, 2001Lynn Cominsky - Cosmology A35045 M-Theory Unification of five different types of superstring theory into one theory called M-theory M-theory has 11 dimensions

47. May 8, 2001Lynn Cominsky - Cosmology A35046 What are the next questions? Can we find the underlying physical principles which have led to us to string theory? Does the correct string (or membrane) theory have 10 or 11 dimensions? Will we ever be able to find evidence for the curled up dimensions? Is string theory really the long-sought “Theory of Everything”? Will any non-physicists ever be able to understand string theory?

48. May 8, 2001Lynn Cominsky - Cosmology A35047 Print Resources Elegant Universe by Brian Greene (Norton) Hyperspace by Michio Kaku (Anchor Books) Fourth Dimension by Rudy Rucker (Houghton Mifflin) Surfing through Hyperspace by Clifford A. Pickover (Oxford)

49. May 8, 2001Lynn Cominsky - Cosmology A35048 Web Resources Visualization of 4 dimensions http://www.cvr.uci.edu/dzmura/4D/default.htm Ned Wright’s Cosmology Tutorial http://www.astro.ucla.edu/~wright/cosmolog.htm http://www.astro.ucla.edu/~wright/cosmolog.htm John Schwarz Superstring pages http://www.theory.caltech.edu/people/jhs/strings/intro.html http://www.theory.caltech.edu/people/jhs/strings/intro.html Patricia Schwarz Official Superstring Theory site: http://superstringtheory.com/index.htm http://superstringtheory.com/index.htm MSNBC http://www.msnbc.com/news/201650.asp

50. May 8, 2001Lynn Cominsky - Cosmology A35049 Web Resources Michio Kaku’s web site http://www.mkaku.org http://www.mkaku.org E. Lowry’s EM Field in Spacetime http://www.ultranet.com/~eslowry/elmag Visualizing tensor fields http://www.nas.nasa.gov/Pubs/TechReports/RelatedPapers/S tanfordTensorFieldVis/CGA93/abstract.html http://www.nas.nasa.gov/Pubs/TechReports/RelatedPapers/S tanfordTensorFieldVis/CGA93/abstract.html Torus and Klein bottle games http://www.northnet.org/weeks/TorusGames/TorusGames.ht ml

51. May 8, 2001Lynn Cominsky - Cosmology A35050 Question of the Week How many dimensions are there? A) 3 B) 4 C) 10 D) 11 E) This question can’t be answered with the available data