A Straight Talk on Curved Space Sencer Yeralan University of Florida

A Straight Talk on Curved Space

Sencer YeralanUniversity of Florida

(C) 2005 www.yeralan.org

GeometryGeometry

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]


Euclid of Alexandria (~ 325-265 BCE)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.


LineLine

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.


Lines, Shapes, and the MindLines, Shapes, and the Mind

Source: S. Lehar (1999), Gestalt Isomorphism and the Quantification of Spatial Perceptionhttp://cns-alumni.bu.edu/pub/slehar/webstuff/isomorph/isomorph.html


Euclidean SpaceEuclidean Space

“Space is Euclidean since it is a modality of the mind…”

Immanuel Kant (1724-1804)


Theorema Egregium (Eminent Theorem)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/)


Transformations and projectionsTransformations 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?


Metric TensorsMetric Tensors

Use a metric tensor to describe “distance” at a given point in space, rather than consider transformed spaces (details).

By the way, the “Reimann Conjecture” is claimed to have been proven by Brenges (source : Purdue).

Riemann (1826-1866)


Absolute Time and Absolute SpaceAbsolute 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)


Michelson-Morley ExperimentMichelson-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.

The most famous experiment that failed.

See the animation.(source : University of Virginia)


On the Thermodynamics of Moving Bodies (1905)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...


Einstein’s EmbankmentEinstein’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?


The Minkowski DiagramThe 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)


Minkowski DiagramMinkowski Diagram

A Minkowski diagram with one space dimension that depicts Einstein’s embankment thought experiment.

(source: www.phy.syr.edu)


Minkowski DiagramsMinkowski Diagrams

Minkowski diagram with one space dimension.

Where exactly is “elsewhere?”

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Date: 10-Nov-2005 Sheet of File: F:\Personal\METU\Philosophy\CurvedSpace\artwork.ddbDrawn By:

Space

Time

Light

"elsewhere"

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Light ConesLight 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)


A Two-Dimensional “Embankment” ExperimentA Two-Dimensional “Embankment” Experiment

Different observers have different perceptions of simultaneity.

“Tilting of the Light Cone”

(picture source: casa.colorado.edu)


Movement of ObjectsMovement 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).

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Coordinate Axes of Other ObserversCoordinate 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 ?

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Objects in MotionObjects 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).

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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?)

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Strange Things Happen when Time is not AbsoluteStrange Things Happen when Time is not Absolute

Spatial Contraction

10 % of c85.6 % of c99 % of c99.99 % c


Time is not AbsoluteTime is not Absolute

How does RED and GREEN view each other’s clocks?

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Strange Things Happen when Time is not AbsoluteStrange 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)


All Reference Points are Equally ValidAll Reference Points are Equally Valid

Time dilation is relative!

“The Light Clock”(picture source: casa.colorado.edu)


Properties of Curved SpaceProperties of Curved Space

Curvature Zero Positive Negative

Parallels One None Many

Sum of angles 180degrees

> 180 degrees

< 180 degrees

Circumference / Diameter

> <


What is the Shape of Space-Time?What is the Shape of Space-Time?

“Since simultaneity does not exists, events cannot be stacked up like pancakes.”

Kurt Godel (1906-1978)

Documents

A Straight Talk on Curved Space Sencer Yeralan University of Florida