1. A Straight Talk on Curved Space Sencer Yeralan University of Florida

2. (C) 2005 www.yeralan.org Geometry Geometry \Ge*om"e*try\, n.; pl. Geometries F. g['e]om['e]trie, L. geometria, fr. Gr. ?, fr. ? to measure land; ge`a, gh^, the earth + ? to measure. So called because one of its earliest and most important applications was to the measurement of the earth's surface. 1. That branch of mathematics which investigates the relations, properties, and measurement of solids, surfaces, lines, and angles; the science which treats of the properties and relations of magnitudes; the science of the relations of space. [1913 Webster]

3. (C) 2005 www.yeralan.org Euclid of Alexandria (~ 325-265 BCE) Every two points lie on exactly one line. Any line segment with given endpoints may be continued in either direction. It is possible to construct a circle with any point as its center and with a radius of any length. If two lines cross such that a pair of adjacent angles are congruent, then each of these angles are also congruent to any other angle formed in the same way. (Parallel Axiom): Given a line L and a point not on L, there is one and only one line which contains the point, and is parallel to L.

4. (C) 2005 www.yeralan.org Line What is a (straight) line? – “Shortest distance” between two points. What is distance? – Non-negative scalar (metric) associated with a pair of points. So is a “line” some distance? – I meant the path that yields the shortest distance. ????? -- Who assigns these “metrics” anyway? – Isn’t it obvious? Get a ruler. How do I know that my ruler will work? – Don’t be obtuse.

5. (C) 2005 www.yeralan.org Lines, Shapes, and the Mind Source: S. Lehar (1999), Gestalt Isomorphism and the Quantification of Spatial Perception http://cns-alumni.bu.edu/pub/slehar/webstuff/isomorph/isomorph.html

6. (C) 2005 www.yeralan.org Euclidean Space “Space is Euclidean since it is a modality of the mind…” Immanuel Kant (1724-1804)

7. (C) 2005 www.yeralan.org Theorema Egregium (Eminent Theorem) Curviture is an intrinsic and local property. Theorema Egregium was not published! Some observations seemingly violate the assumptions of Euclidean space. (see //www.vialattea.net/curvatura/eng/)//www.vialattea.net/curvatura/eng/

8. (C) 2005 www.yeralan.org Transformations and projections How do we map the world? Is underlying space Euclidean? Does space stretch or shrink? Do rulers stretch or shrink? What is a “flat sphere?” Why is the flat sphere isomorphic to the sphere?

9. (C) 2005 www.yeralan.org Metric Tensors Use a metric tensor to describe “distance” at a given point in space, rather than consider transformed spaces (details).details By the way, the “Reimann Conjecture” is claimed to have been proven by Brenges (source : Purdue).provenPurdue Riemann (1826-1866)

10. (C) 2005 www.yeralan.org Absolute Time and Absolute Space Empirical verification is necessary for admissibility of natural phenomena. Metaphysical concepts, e.g., absolute time and absolute space, must be rejected. Thought experiments on, e.g., inertia. Speed as a Mach number – from where did that come? Mach (1838-1916)

11. (C) 2005 www.yeralan.org Michelson-Morley Experiment If light is a wave, then what is the speed and direction of the luminifereous aether that carries it? They built the 1887 “interferometer” at Western Reserve to measure it.interferometer The most famous experiment that failed. See the animation.animation (source : University of Virginia)University of Virginia

12. (C) 2005 www.yeralan.org On the Thermodynamics of Moving Bodies (1905) All reference points are equally valid + The speed of light is the same for all observers = Something must give! Light travels in a “straight” line. Profound influence of Mach and Riemann. The mother of all relativity papers...

13. (C) 2005 www.yeralan.org Einstein’s Embankment Thought experiment (ala Mach?): How do M and M’ deduce when the flashes at A and B occurred? M deduces that the flashes happened simultaneously… M’ insists B happened before A! Who is correct?

14. (C) 2005 www.yeralan.org The Minkowski Diagram "Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." Minkowski (1849-1909)

15. (C) 2005 www.yeralan.org Minkowski Diagram A Minkowski diagram with one space dimension that depicts Einstein’s embankment thought experiment. (source: www.phy.syr.edu)

16. (C) 2005 www.yeralan.org Minkowski Diagrams Minkowski diagram with one space dimension. Where exactly is “elsewhere?”

17. (C) 2005 www.yeralan.org Light Cones A Minkowski diagram with two spatial dimensions: a flash at the center is reflected off the mirrored walls, equidistant from the center. In our experience, the cone should be four dimensional, with three spatial and one time dimension. What is “outside” the cone? Do we perceive it? Is it meaningful to even consider it? (source:casa.colorado.edu)casa.colorado.edu

18. (C) 2005 www.yeralan.org A Two-Dimensional “Embankment” Experiment Different observers have different perceptions of simultaneity. “Tilting of the Light Cone” (picture source: casa.colorado.edu)casa.colorado.edu

19. (C) 2005 www.yeralan.org Movement of Objects A common object has length. Each point of the object traces a curve in the space-time continuum. Here, we trace the leading and trailing edges of the spaceship (its bumpers).

20. (C) 2005 www.yeralan.org Coordinate Axes of Other Observers How does the observer at the embankment see the time and space axes of the observer on the train? Why is  equal to  ?

21. (C) 2005 www.yeralan.org Objects in Motion Consider two spaceships traveling in opposite directions with respect to an observer on Earth. The space-time events A and B denote when the front and back bumpers of the spaceships pass each other (our viewpoint).

22. (C) 2005 www.yeralan.org Look at the lines of simultaneity… How does Green and Red judge each other’s length? (Hint: would Red observe event A or event B to occur first?)

23. (C) 2005 www.yeralan.org Strange Things Happen when Time is not Absolute Spatial Contraction 10 % of c 85.6 % of c 99 % of c 99.99 % c

24. (C) 2005 www.yeralan.org Time is not Absolute How does RED and GREEN view each other’s clocks?

25. (C) 2005 www.yeralan.org Strange Things Happen when Time is not Absolute Time Dilation Different observes measure clocks running at different speeds. Any clock that is in motion with respect to an observer slows down. This leads to the well- known “twin paradox.” “The Light Clock” (picture source: casa.colorado.edu)casa.colorado.edu

26. (C) 2005 www.yeralan.org All Reference Points are Equally Valid Time dilation is relative! “The Light Clock” (picture source: casa.colorado.edu)casa.colorado.edu

27. (C) 2005 www.yeralan.org Properties of Curved Space CurvatureZeroPositiveNegative ParallelsOneNoneMany Sum of angles180 degrees > 180 degrees < 180 degrees Circumference / Diameter  >  < 

28. (C) 2005 www.yeralan.org What is the Shape of Space-Time? “Since simultaneity does not exists, events cannot be stacked up like pancakes.” Kurt Godel (1906-1978)