A Straight Talk on Curved Space Sencer Yeralan University of Florida

(C) 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]

(C) Euclid of Alexandria (~ 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.

(C) 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.

(C) Lines, Shapes, and the Mind Source: S. Lehar (1999), Gestalt Isomorphism and the Quantification of Spatial Perception

(C) Euclidean Space “Space is Euclidean since it is a modality of the mind…” Immanuel Kant ( )

(C) 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 //

(C) 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?

(C) 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 ( )

(C) 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 ( )

(C) 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

(C) 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...

(C) 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?

(C) 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 ( )

(C) Minkowski Diagram A Minkowski diagram with one space dimension that depicts Einstein’s embankment thought experiment. (source:

(C) Minkowski Diagrams Minkowski diagram with one space dimension. Where exactly is “elsewhere?”

(C) 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

(C) 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

(C) 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).

(C) 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  ?

(C) 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).

(C) 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?)

(C) Strange Things Happen when Time is not Absolute Spatial Contraction 10 % of c 85.6 % of c 99 % of c % c

(C) Time is not Absolute How does RED and GREEN view each other’s clocks?

(C) 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

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

(C) Properties of Curved Space CurvatureZeroPositiveNegative ParallelsOneNoneMany Sum of angles180 degrees > 180 degrees < 180 degrees Circumference / Diameter  >  < 

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