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Dimensional analysis, quantum gravity and the
electromagnetic spectrum
At
the beginning of the twenty first century, there is much research into the
ideas of quantised gravitational action. It is argued that the ideas of relativity
and quantum theory will combine to produce a greater understanding of the
physical universe. Some say that this understanding will be complete! Of the
two, quantum theory has been tested more successfully hitherto but high
precision experiments on the predictions of relativity theory are close to
yielding results. Primarily, quantum gravity is discussed mathematically yet
much of the publicly accessible material includes little of the mathematics to
explore. This note describes pedagogically how dimensional analysis is used and
shows how relativity theory and quantum mechanics combine through knowledge of
their fundamental constants using the technique.
The
idea that the speed of light in vacuo is
dependent upon photon energy is discussed. The dependency (photon speed falls
as its energy rises) is reformulated here in terms of photon frequency and a
forecast that the electromagnetic spectrum has a high frequency cut-off is
presented. A recent experimental test failure of the ideas is discussed that
highlights the required accuracy in such trials.
Dimensional Analysis
Dimensional analysis is a mathematical technique. It allows for the physical understanding of phenomena through relationships between the units of measure (or dimensions) employed. In everyday existence we meet dimensions of length [metres, feet…], mass [kilogrammes, pounds…], time [seconds, hours…], temperature [Kelvin, Celsius, Fahrenheit… degrees] and (less commonly) electric charge [Coulombs].
Notes:
1)
Square brackets, [ ], indicate “units of”.
2)
(a)1/2, or a to the power one-half, is the
square-root of a.
3) Tilda (~) indicates approximate equality.
There
are many and varied systems of units including the Systčme
International (SI) that is widely adopted. The SI is based upon the following
properties: the amount of substance or mole [mol]; length in metres [m]; mass
in kilogrammes [kg]; time in seconds [s]; the Kelvin
degree of hotness [K]; the radian of angular measure [c]; the
Coulomb of electric charge [C] and the candela [cd]
of luminous intensity. This system, therefore, requires 9 dimensions to specify
an object uniquely: the amount of substance within; its spatial location (three
perpendicular distances or one distance and two angles from an origin); its
mass; the time of observation; how hot it is; how much electric charge it
possesses and how shiny it is.
The
SI describes object properties but some processes require additional
dimensional considerations: the description of radioactive decay requires a
statistical dimension and Higgs’ analysis includes a term that is dependent
upon the orientation of fields. The class of ideas known as M-theory adds
another space-like dimension to the litany and describes an 11 dimensional
space-time with six of its space-like dimensions compactified.
To define the compactification of spatial dimensions,
imagine a tube (e.g. a hosepipe or drinking straw) seen from a distance that is
large compared to its outside diameter or study the following illustration:
Compactification
A tube is represented by a
sequence of overlapping circles.
The tube diameter shrinks to
1-D line of dots.
Two spatial dimensions (the
width and height of the tube) are compactified
(curled into small regions) but the tube length remains fixed.
Image source: http://www.vcnet.com
The
speed, U, of an object is a combination of unit measure. Speed is defined as
the distance travelled in a unit of time, SI speed is measured in metres per
second, [U] = [ms-1]. Other unit systems have speed in miles per
hour [mph] or, less usefully, furlongs per fortnight. Since, 1 mph is
equivalent to moving 1609 metres in 3600 seconds then 1mph is approximately the
same as one half of 1ms-1.
When
like is compared with like the relationships between various systems of units
are purely multiplicative.
Example:
Estimate ‘c’, the speed of light in vacuo, in
units of furlongs per fortnight.
c
~ 2.99x108 metres per second; 1 furlong ~ 201 metres; 1 fortnight =
14 x 24 x 60 x 60 seconds.
c
[furlongs per fortnight] = c [metres per second] x (furlongs per metre) x
(seconds per fortnight).
Hence, c ~ 1.8x1012 furlongs per fortnight.
The
linear acceleration, a, of an object is the rate of change of its speed:
[a] = [speed per unit time] = [metres per second per second]
= [ms-2].
In
seventeenth-century Italy, Galileo Galilei noticed
that pendulum length and time to swing were closely related. Further study
showed that the period T [s] of a simple pendulum depends upon its length l [m]
and the acceleration due to gravity g [ms-2] but not upon the
mass of the bob. T is proportional to a function of l and g alone.
Q:
What is the functional dependence of T on l and g?
A:
Using dimensional analysis:
The
period, T, has the dimension of time [s].
The
required function could be the product (lg), the
ratio (g/l), the ratio (l/g) or some other combination...
The
quantity lg: [lg] = [m ms-2]
=[m2s-2]
The
quantity g/l: [g/l] = [ms-2/m] = [s-2],
The
quantity l/g: [l/g] = [m/ms-2] = [s2].
Hence,
the period of a pendulum will be proportional to (l/g)1/2.
Dimensional
analysis will not yield the complete relationship since there is no way to
determine the magnitude of any dimensionless factors. The actual period of a
pendulum bob is, T = 2π(l/g)1/2. The
factor of 2π is not found by dimensional analysis.
Air
molecules are small in comparison with the wavelength of visible light: the
diameter of a nitrogen or oxygen molecule is around 10-10m whilst
the wavelength of visible light spans approximately 0.5 to 1μm (1 μm=10-6m). When light is scattered by small
objects the process is known as Rayleigh Scattering (reference 1). The intensity, I, of solar
radiation scattered by the molecules of Earth’s atmosphere is functionally
dependent upon the wavelength, λ, of the incident radiation, I0.
The scattering dipole radiates in all directions and hence the intensity
follows an inverse square law with distance, r, from the site of scatter. The
field of the dipole is proportional to the dipole moment and for a uniform
exciting field this, in turn, is proportional to the volume, V, of the
particle. The intensity is proportional to the square of the field and hence V2.
Any dependency upon the refractive indices of the particle and medium, n1
and n2, has no effect in the dimensional analysis since these
themselves are dimensionless quantities.
The
functional relationship for scattered radiation is:
I =
f (n1, n2) (V2 r--2 λb) I0
The
quantity (V2 r--2 λb)
is, therefore, dimensionless:
[V2
r--2 λb] = [m6
m-2 mb]
Hence
b = -4 and this inverse fourth power dependency upon wavelength is why clear
skies are blue.
Energy
is defined as the capacity of an object to do work. The SI unit of energy is
named the Joule [J].
The
kinetic energy of an object of mass m travelling at speed U is given by Ekinetic = ˝mU2.
A
stationary object above the ground has the potential to fall in the
gravitational field. The gravitational potential energy of mass m at a height h
above a reference in a region of gravitational acceleration g is given by Epotential = mgh.
In
the SI:
[Ekinetic] = [kilogramme metre squared per second
squared] = [kg m2 s-2] = [J].
[Epotential] = [kilograme
metre per second squared metre] = [kg m2 s-2] = [J].
The
constants of Einstein’s relativity theory include the speed of light, c, and
Newton’s constant of gravitational attraction, G. The dimensions of c are those
of speed [ms-1], whilst [G] may be found from Newton’s second law of
motion and the Universal Law of Gravitation:
For
a fixed mass, m, Newton’s second law of motion can be written as F=ma
[F]
= [Newtons] = [kilogramme metre per second squared] =
[kg m s-2]
The
Universal Law of Gravitation may be rewritten as G=F r2/(m1 m2)
[G]
= [Newton metre squared per kilogramme squared] = [m3 kg-1
s-2].
Work
in quantum mechanics began with the quantisation of radiant energy: E = hν = hω/2π where h is Planck’s constant,
ν (nu) is the frequency of the radiation in
hertz [s-1] and ω (omega) is the angular frequency in radians
per second.
[h]
= [E ν-1] = [Joule seconds] = [kg m2 s-1].
The
constants c, G and h can be arranged in different ways and three particular
arrangements yield quantities with the dimensions of length, time and
mass/energy respectively.
Dimensional
analysis of the quantity (G h/c3):
[G
h/c3] = [m3 kg-1 s-2 x kg m2
s-1 x m-3 s3] = [m2]
The
square root, Lpl = (G h/c3)1/2,
has the dimension of length [m] and is known as the Planck length.
Similarly,
the quantity Tpl = (G h/c5)1/2
has the dimension of time [s].
Thirdly,
Mpl = (hc/G)1/2
has the dimension of mass [kg].
Relativity
theory shows that mass and energy are related (E=mc2) and Mpl is often presented in terms of its
equivalent energy and the preferred unit of energy is the electron volt [eV] (1 eV = 1.6x10-19
J).
Numerical
values of Lpl, Tpl
and Mpl are shown in the appendix.
An
early example where the Planck length appears is Hawking’s
work on the entropy, S, of black holes. He shows that the entropy is
proportional to the area, A, of the event horizon (ref 2). The constant of
proportionality includes a contribution from the Planck length squared and Boltzmann’s constant, k:
S =
A k c3 / (4 G h) = A k / (4 Lpl2) Joules per
Kelvin degree [J K-1].
That
the speed of light in vacuo has a dependency
on the energy of the photon is a relatively recent and novel idea. The idea
appears counterintuitive: increase a photon’s energy and its speed reduces.
What is going on? The resolution leads to the prediction of a high frequency
cut-off in the electromagnetic (e-m) spectrum. This upper limit to the
frequency of e-m radiation is a consequence of the quantisation of the geometry
of space-time. Einstein’s postulate that the speed of light is a constant of
inertial reference is refined only at enormous energies.
No
upper limit to the frequency of e-m radiation was envisaged hitherto, Lorrain & Corson (ref 3), at high frequencies the e-m spectrum is often indicated as
an arrow to indicate that gamma-ray (γ-ray) frequencies increase into
undefined regions. Camelia et al (ref 4), present a
three-momentum-dependency of the speed of light that can be identified with a
similar result in Camelia et al (ref 5). The
dependency is such that high-energy photons propagate through the quantised
metric of space-time more slowly than low-energy ones. In absolute terms the
dependency is miniscule for most of the e-m spectrum but increases markedly as
photon energies rise beyond 1013 TeV (1 TeV = 1012 eV) or more
than 7 megajoules per photon. The appendix
reformulates the relationship between a photon’s actual speed,
c’, and the more familiar c of Maxwell’s equations as:
c’/c
= 1 – (ν Tpl) where ν is photon
frequency and Tpl is the Planck time.
This
is shown in Figure 1 and predicts a maximum frequency, νmax=1/Tpl
=7.40x1042 Hertz, that electromagnetic radiation can attain in vacuo. It is noted that this frequency is about twelve
orders of magnitude beyond the highest frequency of γ-photons measured
hitherto: νmax(observed)~1030 Hz (Emax(observed)~0.7 millijoules~4000 TeV).
Experiments to test
the notion that blue travels slower than red across a vacuum probably remain
some time ahead and are not trivial. In an early
experiment of this kind, Philip Kaaret (ref 6)
reports that no quantum gravity effect is discernible in the arrival times of
distinct γ-ray groups emitted from within the Crab Nebula. He notes,
however, that the studied regime is about an order of magnitude shy of ‘where
things will happen’ as predicted by some quantum gravity workers e.g. Ed Witten (ref 7). This is confirmed in appendix point 2: the
technique is sound but the required measurement precision is beyond the current
experiment by more than an order of magnitude. Kaaret
also warns that single source astronomical emissions may prove insufficient to
determine the veracity of the claim: he notes that variations in source
emissions may overwhelm those sought at the observatory. Lee Smolin (ref 8) suggests that project GLAST (the Gamma Ray
Large Area Telescope, due to launch in 2006) should permit more rigorous tests
of the ideas.
Some
of the constants of relativity theory and quantum mechanics can be combined
mathematically to suggest limits to space, time and mass/energy through a
technique known as dimensional analysis.
It
has been shown theoretically and recently that the speed of light in vacuo depends on the geometry of space-time and that
the dependency can be expressed in terms of photon energy: higher energy
photons travel slower. Reformulation of this dependency in terms of photon
frequency leads to the prediction that e-m radiation with a frequency greater
than a maximum value will not traverse a vacuum. The maximum value for the
frequency of e-m radiation is calculated to be 7.40x1042 Hertz.
The
rate of technological advance as measured by the clock-speed of microprocessor
devices suggests that frequencies near this magnitude will be reached in the
not-too-distant future. In the early 1980s many Zilog™
devices operated at 4 MHz, in the early years of the new millennium, Intel™
offered 4 GHz devices. This 1000-fold increase in 20 years extrapolated
linearly to 1042 Hz will take another 220 years. Linear
technological advance, however, is only modest compared with
1:
The steps from Camelia et al (ref 4) equation 2.16b
to the dependency presented here are as follows:
V=
c - |P|/κ + O(1/κ2) is their
equation 2.16b where V is photon speed, P is momentum and κ is the Planck
mass.
Firstly,
O(1/κ2) is the sum of terms from a
power-series expansion and is neglected as being small.
Secondly,
c’ is used here instead of V to give c’/c = 1 – P/cκ.
Next,
P=E/c and E=hν are used to give c’/c = 1 –
hν/c2κ.
Finally,
κ = (hc/G)1/2
and Tpl = (hG/c5)1/2
are used to give the required expression: c’/c = 1 – (ν Tpl)
Strictly,
this indicates that c’ becomes negative for frequencies above νmax = 1/Tpl. Negative c’ may be
interpreted as the radiant energy travelling backwards through time.
2:
To calculate the time of flight difference, ΔT s, between 0.1 and 2.0 GeV photons from the Crab Nebula at a distance, d, of 22000
parsecs:
Photons
of these energies travel at speeds c’1 and c’2
respectively and the difference in speed is Δc’ ms-1.
ΔT
= Δc’d/(c1’c2’)
s.
Δc’ = cTplΔE/h ms-1.
ΔE
= 1.9 GeV ~ 3x10-10 J.
Δc’ = 1.4x10-11 ms-1.
d =
2.2x103 x 3.1x1016 ~ 7x1019 m.
ΔT ~ 1.4x10-11 x 7x1019/9x1016 ~
10ns.
Philip
Kaaret (ref 6) has the detectable limit of ΔT in
his experiment at 0.35 milliseconds (350 ns): a factor of 35 from the
requirement.
3:
The values of physical constants are known to very high precision. Here they
are rounded to:
c =
2.99x108 m s-1 The speed of light in vacuo
G =
6.67x10-11 m3 kg-1 s-2 The universal constant of gravitation
h =
6.63x10-34 J s [kg m2 s-1] Planck’s constant
k =
1.38x10-23 JK-1
Boltzmann’s
constant
Lpl = 4.05x10-35 m
Tpl = 1.35x10-43 s
Mpl = 5.46x10-8 kg
Epl = 4.90x109 J = 3.06x1028 eV
1 eV = 1.6x10-19 J
1 H.C. van der Hulst,
1957, “Light scattering by small particles”, John Wiley & Sons Inc.,
2
Jack Sarfatti, http://www.stardrive.org/math3/hawking1.htm
3
Paul Lorrain and Dale Corson,
1970, “Electromagnetic Fields and waves”, second edition, WH Freeman & Co,
4
Giovanni Amelino-Camelia, Jerzy
Lukierski and Anatol Nowicki, 1997, “k-DEFORMED COVARIANT PHASE SPACE AND
QUANTUM-GRAVITY UNCERTAINTY RELATIONS”, hep-th/9706031: http://arxiv.org/PS_cache/hep-th/pdf/9706/9706031.pdf
5 G
Amelino-Camelia, J Ellis, NE Mavromatos,
VD Nanopolous and S Sakar,
1998, Nature, 393, 763
6 Philip Kaaret, 2003, “Pulsar Radiation and
Quantum Gravity”, Astronomy and Astrophysics, July. http://arxiv.org/PS_cache/astro-ph/pdf/9903/9903464.pdf
7 Ed
8
Lee Smolin, 2003, “Loop Quantum Gravity”, http://www.edge.org/3rd_culture/smolin03/smolin03_index.html
contact: Ian Consterdine