Space-time Invariance
and Quantum Gravity
By Dr. Harold Williams
of Montgomery College Planetarium
Space-time Invariance and Gravity: only c and G
General Relativity 1915, continuous Real Numbers
Dr. Harold Alden Williams, Montgomery College Planetarium at the
Takoma Park/Silver Spring Campus
. Space-time Invariance and Quantum Gravity: or
how c, G, and h create the fabric of time-space
(reality)!
Dr. Harold Alden Williams, Montgomery College Planetarium at the
Takoma Park/Silver Spring Campus
World Lines
• t for time
• x for distance
• dx/dt speed
Space-time
• At least one
space
dimension not
shown
World lines, sheets, and volumes
of 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.
v 1  v2
v 1 v2 
2
1  v 1 v2 c
v is the speed, this formula actually is
only correct when v1 and v2 are in the same
direction, velocity addition formulae.
Speed of light, fastest speed
• c=by definition=299795458 (exact) m/s
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.
G, Newtonian constant of gravitation
G=6.673(10) × 10-11 m3 kg-1 s-2
measured
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
2GM
Rs  2
c
Rs Schwartzshield Radius
G Newtonian Gravitational Constant
M Mass causing curvature
c speed of light
Einstein’s Energy Mass
relationship
E  Mc  M  E / c
2
2
E is the energy, M is the
mass, and c the speed of light
Speed = frequency wavelength
tautology
c   f  f  c  or   c / f
c is the wave speed, speed of light in a
vacuum here.
λ is the wavelength
f is the frequency
Planck’s Law
Quantum Mechanics
E  hf  E  hc 
h is Planck’s constant, energy time
h, Planck constant
• h=6.62606876(52)×10-34 J s (measured)
• J=Joules = kg m2 s-2 (Energy SI unit)
• ħ=h/(2π)=1.054571596(82)×10-34 J s
some times called the Dirac Constant
Substituting things and solving for
the wavelength
2GE / c
2G hc  / c
  Rs  2    Rs 
2
c
c
2
1/ 2
2Gh
 2Gh 
  3   3 
c
 c 
2
2
Substituting things and solving for
the frequency
2
2GE / c
2Ghf / c
  c / f  Rs 

2
2
c
c
f
2
1/ 2
 c 
c

 f 

2Gh
 2Gh 
5
5
1/ 2
 2Gh 
f  1/ T  T  1/ f   5 
 c 
T is the period of oscillation
2
(Gh/(2c5))1/2=5.3905x10-44stP the
Planck time, a chronon, smallest time,
quanta of time, when! Time is quantitized
at this level, so called “third
quantitization.”
(Gh/(2c3))1/2=1.6160x10-35mlP the
Planck length, an extention, smallest
length, quanta of distance, where! Space
is quantitized at this level, so called “third
quantitization.”
History of Planck length, time, and mass
In 1899 a year before he formulated
blackbody radiation in December of 1900
in Prussian Academy of Sciences. Max
Planck: 'Über irreversible
Strahlungsvorgänge'. Sitzungsberichte
der Preußischen Akademie der
Wissenschaften, vol. 5, p. 440-479-480
(1899).
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/(2G))1/2=1.95x109 joules=eP the
Planck Energy
eP=1.28x(1028 eV or 1019GeV or 1016TeV
e0=0.467 tons of TNT
Planck mass
eP/c2=mP=2.17x10-8 kg
Meaning of the Planck Mass, mP
• me = 9.10938188(72) × 10-31 kg
• Σ i=1 to 3 Δmνi = 0.7eV= 1.2 × 10-36 kg
reference
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 Relativity
the Special and General Theories for the Structure of the Universe
by Moshe Carmeli (the man)
• 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)
Relativity: Modern Large-Scale
Spacetime Structure of the Cosmos
•
•
•
•
•
•
Moshe Carmeli editor
Probably his last book
Finished by John Hartnett
Preface finished by Moshe Carmeli
Also in this text Gianluca Gemelli
5-dimensional special relativistic
hydrodynamics and cosmology by Gianluca
Gemelli
End for now!