GENERAL RELATIVITY

A GRAND TOUR OF PHYSICS

DR. GEORGE DERISE PROFESSOR EMERITUS, MATHEMATICS THOMAS NELSON COMMUNITY COLLEGE SPRING 2019

APR. 19, 2019 1:30 – 3:30 TNCC ROOM 328.

LECTURE 5

A QUICK SURVEY OF NON-EUCLIDEAN GEOMETRY

PARALLEL POSTULATE EUCLIDEAN GEOMETRY: Through a given point P, not on a given line L, there is one and only one line that can be drawn through P parallel to L.

MODEL OF A GAUSS-BOLYAI-LOBATCHEVSKI NON EUCLIDEAN GEOMETRY: ABSTRACT GBL GEOMETRY MODEL ENTIRE PLANE SURFACE OF A SADDLE POINT POINT ON THE SADDLE LINE GEODESIC OF THE SADDLE GBL PARALLEL POSTULATE THROUGH A GIVEN POINT P, NOT ON A GIVEN LINE l, THERE ARE AT LEAST TWO LINES THROUGH P PARALLEL TO l .

MODEL OF A GAUSS-BOLYAI-LOBATCHEVSKI NON EUCLIDEAN GEOMETRY: ABSTRACT GBL GEOMETRY MODEL ENTIRE PLANE SURFACE OF A PSEUDOSPHERE POINT POINT ON THE PSEUDOSPHERE LINE GEODESIC OF THE PSEUDOSPHERE GBL PARALLEL POSTULATE HOLDS!

MODEL OF A RIEMANNIAN NON EUCLIDEAN GEOMETRY:

BERNHARD RIEMANN

1826-1866

KARL FRIEDRICH GAUSS

86o 13' 58.366''

53o 6' 45.642''

40o 39' 30.165''

180o 00' 14.173''

Can a bug living on a two dimensional surface determine the geometry of the surface?

INTUITIVE IDEA OF CURVATURE CURVES IN THE PLANE

K=0

INTUITIVE IDEA OF CURVATURE 2 DIMENSIONAL SURFACES

Why are we doing this math?

IS THERE A “FORCE OF ATTRACTION” BETWEEN A AND B ?

GENERAL RELATIVITY - “THERE IS NO FORCE” -IT’S JUST THE GEOMETRY

APPLE’S SURFACE- RIEMANNIAN GEOMETRY MODEL OF SPACETIME

ANT GOING ALONG A GEODESIC- WORLD LINE THROUGH SPACETIME OF A FREE PARTICLE

STEM OF APPLE- BENDING OF SPACETIME BY A MASS

LOCALLY THE APPLE IS FLAT

EQUIVALENCE PRINCIPLE

There is no experiment that will discern the difference between the effect of gravity and the effect of acceleration.

GRAVITATIONAL MASS ~ INERTIAL MASS

...I know now that if I break my neck by falling off a cliff, my death is not to be blamed on the force of gravity (what does not exist is necessarily guiltless), but on the fact that I did not maintain the first curvature of my world-line, exchanging its security for a dangerous geodesic. Relativity- the General Theory J.L. Synge

CONNECTING THE EQUIVALENCE PRINCIPLE WITH THE GEOMETRY OF SPACETIME: THE WORLD LINE OF THE EARTH AROUND THE SUN IS A GEODESIC IN 4 DIMENSIONAL SPACETIME- I.E. THE SHORTEST 4 DIMENSIONAL SPACETIME DISTANCE BETWEEN TWO POINTS. (GEOMETRY). NEWTON: FORCE = MASS x ACCELERATION (PHYSICS)

MATTER TELLS SPACE HOW TO CURVE, AND CURVED SPACE TELLS MATTER HOW TO MOVE.

JOHN WHEELER

GENERAL RELATIVITY

THE CURVATURE OF THE METRIC 𝒈𝝁𝝂 is related to

THE MATTER-ENERGY DISTRIBUTION OF SPACETIME

by

EINSTEIN’S FIELD EQUATION

𝑮𝝁𝝂  =  𝟖𝝅𝑻𝝁𝝂

CURVATURE OF SPACETIME ~ ENERGY DENSITY OF MATTER

GEOMETRY ~ PHYSICS

SPACETIME: 4 DIMENSIONAL RIEMANNIAN MANIFOLD

HILBERT GOT EINSTEIN’S FIELD EQUATIONS BY USING AN ACTION PRINCIPLE 5 DAYS BEFORE EINSTEIN

GENERAL RELATIVITY EXPERIMENTAL VERIFICATION SOLAR ECLIPSE EXPEDITION -1919

GRAVITATIONAL LENSING: The gravitational field of a massive object causes light rays passing close to that object to be bent. Mass bends light.

PRECESSION OF THE PLANET MERCURY

POUND-REBKA EXPERIMENT 1960 HARVARD PHYSICS TOWER

Top-bottom difference: 1 second in 100 million years

verified Einstein’s 1911 prediction that gravity could change light’s frequency.

GRAVITATIONAL REDSHIFT

GPS

accuracy of 5 to 10 meters

24 satellites with atomic clocks

Special Relativity predicts clocks on the satellites fall behind clocks on the ground by 7 microseconds per day

General Relativity predicts clocks on the satellites get ahead of clocks on the ground by 45 microseconds per day.

combining these two relativistic effects:

the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)

PHYSICS = GEOMETRY

GRAVITATIONAL WAVES: Disturbances in the curvature (fabric) of spacetime generated by accelerated masses that propagate as waves outward from their source at the speed of light. First proposed by Henri Poincaré in 1905 Predicted by Einstein Theory of General Relativity-1916. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation.

LIGO: The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory to detect cosmic gravitational waves. The first direct observation of gravitational waves - September 2015. The waves given reached Earth as a ripple in spacetime that changed the length of a 4-km LIGO arm by a ten thousandth of the width of a proton (proportionally equivalent to changing the distance to the nearest star outside the Solar System by one hair's width.)

Far field solution of Einstein’s Field Equation The Minkowski space-time of Special Relativity

Einstein’s Field Equation on a wall of Museum Boerhaave, Leiden

Gravitational lensing phenomena-deflection of light by an intervening mass

Karl Schwarzschild (1873–1916)

SCHWARZSCHILD BLACK HOLE SOLUTION

SCHWARZSCHILD METRIC VALID SOLUTION OF THE EINSTEIN FIELD EQUATIONS

BLACK HOLES A hole in space-time Black: even light can’t escape

SUN 2

23KM

GMr

c

Density= 20,000million tons/cubic cm

NO HAIR THEOREM: John Wheeler “ A Black Hole has no hair.” A black hole has only three characteristics (1) mass (2) angular momentum (3) charge

WILL THE UNIVERSE EXPAND FOREVER?

A CYCLIC UNIVERSE

NO BIG BANG-BIG CRUNCH SINGULARITIES

HARMONIC OSCILLATING UNIVERSE

t

R(t) CYCLIC UNIVERSE

41

Cosmic microwave background COSMIC MICROWAVE BACKROUND RADIATION (CMBR)

Remnant Electromagnetic radiation from the Big Bang first predicted -1948; first observed -1965 NASA's Cosmic Background Explorer (COBE): 1989 - 1993 NASA-PRINCETON’S Wilkinson Microwave Anisotropy Probe (WMAP): 2003 - 2012 EUROPEAN SPACE AGENCY (ESA)’s Planck Space Observatory 2009 - 2013

COBE: Black-body curve of CMB First “baby pictures” of the Universe Intrinsic anisotropy of CMB-detecting early galaxies

WMAP: Mapped the pattern of tiny fluctuations in the CMB radiation (the oldest light in the Universe) Produced the first fine-resolution (0.2 degree) full-sky map of the microwave sky Measured the fluctuations of density in the early universe that produced the first galaxies

PLANCK SPACE OBSERVATORY: Mapped the anisotropies of the CMB at microwave and infra-red frequencies Improved on WMAP Confirmation of the Universe having a 26% content of dark matter Validation of the simplest models of inflation

Determined the universe to be 13.77 billion years old to within a half percent Nailed down the curvature of space to within 0.4% of "flat" Euclidean

Determined that ordinary atoms (baryons) make up only 4.6% of the universe Completed a census of the Universe -dark matter (matter not made up of atoms) is 24.0% Dark energy, (cosmological const.) makes up 71.4% of the Universe

WHY INFLATION? PROBLEMS WITH THE STANDARD BIG BANG

WHY IS OUR UNIVERSE SO HOMOGENEOUS? (BETTER THAN 1:10,000)

WHY IS IT ISOTROPIC?

WHY ALL OF ITS PARTS STARTED EXPANDING SIMULTANEOUSLY?

WHY IS IT FLAT?

INFLATION MAKES THE UNIVERSE FLAT, HOMOGENEOUS AND ISOTROPIC

IN THIS SIMPLE MODEL THE UNIVERSE TYPICALLY GROWS 101000000000000 TIMES DURING INFLATION.

HAWKING RADIATION (1974)

BLACK HOLES ARE NOT ENTIRELY BLACK BUT EMIT SMALL AMOUNTS OF THERMAL RADIATION

BEKENSTEIN-HAWKING ENTROPY FORMULA

S=A

𝟒

SU(3) x SU(2) x U(1) STANDARD MODEL OF PARTICLE PHYSICS