Space-time Invariance and Quantum Gravity

By Dr. Harold Williams

of Montgomery College Planetarium

World Lines

• t for time

• x for distance

• dx/dt speed

Space-time

• At least one space dimension not shown

World lines, sheets, and volumesof particles, strings, and branes

• We know we live in a universe with at least 3 space dimensions that we have some freedom in.

• Maybe more elaborate structures exist, but we do not as yet perceive them.

Special Relativity

• http://en.wikipedia.org/wiki/Special_relativity • http://en.wikipedia.org/wiki/Special_relativity_for_beginners

• “Nothing but Relativity” by Palash B. Pal shows that only Galilean and Einstein Relativity are even possible with the “Principle of Relativity”

• “Principle of Relativity” ≡ “Physical laws appear the same to any inertial observer.”

There is a Fastest Speed• What ever that fastest speed is nothing can move

faster than fastest. c is the fastest speed. Experimentally this, c, is the speed of light, electromagnetic radiation, in a vacuum.

1 21 2 2

1 21

v vv v

v v c

v is the speed, this formula actually is only correct when v1 and v2 are in the same direction, velocity addition formulae.

Correct velocity addition formulae for non-colinear velocities

http://en.wikipedia.org/wiki/Addition_of_Velocities_Formula

Newton’s Universal Law of Gravity

F12 is the force on object 2 due to object 1. G is the gravitational constant. m1 and m2 are respectively the masses of objects 1 and 2. r21 = |r2 − r1| is the distance between objects 2 and 1.

is the unit vector from object 1 to 2.

Einstein’s General Theory of Relativity, a theory of gravity

Spherically Symmetric Vacuum Solution to General Relativity

• Invariant ds2 in spherical coordinates

Black Hole Radius, Schwartzshield Radius

2

2s

GMR

c

Rs Schwartzshield Radius

G Newtonian Gravitational Constant

M Mass causing curvature

c speed of light

Einstein’s Energy Mass relationship

2 2/E Mc M E c

E is the energy, M is the mass, and c the speed of light

Speed = frequency wavelength tautology

or /c f f c c f

λ is the wavelength

f is the frequency

Planck’s Law

E hf E hc h is Planck’s constant, energy time

Substituting things and solving for the wavelength

2 2

2 2

2 / 2 /s s

GE c G hc cR R

c c

1/ 22

3 3

2 2Gh Gh

c c

Substituting things and solving for the frequency

2 2

2 2

2 / 2 // s

GE c Ghf cc f R

c c

1/ 25 52

2 2

c cf f

Gh Gh

1/ 2

5

21/ 1/

Ghf T T f

c

T is the period of oscillation

(Gh/(2c5))1/2=5.3905x10-44stP the Planck time, a chronon, smallest time,

quanta of time, when! Time is quantitized at this level, so called “third

quantitization.”(Gh/(2c3))1/2=1.6160x10-35mlP the Planck length, an extention, smallest

length, quanta of distance, where! Space is quantitized at this level, so called “third

quantitization.”

46x109 light-years radius of the universe, so an estimate of the current information content, pixels, of the universe, since creation could be made. So here it is the current estimate of the information content of the universe, which is an always increasing number, because time is advancing; we get older and not younger; it’s a one way trip; no time travel allowed (you can not kill your mother before you were conceived); and the observable universe is expanding and accelerating since the creation event: 13.7x109 years/tP*4/3*(46x109 light-years/lP)3 =6.6x10245.

Wikipedia Resources

• Cosmic Background Radiation

• Quantum Gravity

• General Relativity

• Planck Time

• Planck Units

Wikipedia Cosmological Resources

• Cosmic Inflation

• Cosmological Constant

(hc5/(2G))1/2=1.95x109 joules=eP the Planck Energy

eP=1.28x(1028 eV or 1019GeV or 1016TeVe0=0.467 tons of TNT

eP/c2=m0=2.17x10-8 kg=2.17x10-5gm the Planck mass

End

Sort of

Hubble Parameter

• H0 ≈ 70 km/sec 1/Mpc =2.3x10-18 sec-1

• H0 ≈1/4.4x1017sec=1/13.9x109years

• 1/ H0 is the age of the universe (somewhat model dependent)

• H0 ≈10-33eV

Cosmological Relativitythe Special and General Theories for the Structure of the Universe

by Moshe Carmeli• ds2=dx2+dy2+dz2-c2dt2-τ2dυ2

• x, y, z usual meaning lengths in space

• c, maximum speed in the universe, speed of light in a vacuum

• t, time, what good clocks keep

• τ, age of the universe=1/ H0

• υ, recessional velocity (of the galaxy)