In Pursuit of the Unknown (62 page)

where
u
(
x, t
) is the vertical position of location
x
on the string at time
t.

2
For an animation see
http://en.wikipedia.org/wiki/Wave_equation

3
In symbols, the solutions are precisely the expressions

u
(
x, t
) =
f
(
x
−
ct
) +
g
(
x
+
ct
)

for any functions
f
and
g
.

4
Animations of the first few normal modes of a circular drum can be found at
http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum

Circular and rectangular drum animations are at
http://www.mobiusilearn.com/viewcasestudies.aspx?id=2432

1
Suppose that
u
(
x, t
) = e
^{−n
2
αt}
sin
nx
. Then

Therefore
u
(
x, t
) satisfies the heat equation.

2
This is JFIF encoding, used for the web. EXIF coding, for cameras, also includes ‘metadata' describing the camera settings, such as date, time, and exposure.

1
http://www.nasa.gov/topics/earth/features/2010-warmest-year.html

1
Donald McDonald. How does a cat fall on its feet?, New Scientist
7
no. 189 (1960) 1647–9. See also
http://en.wikipedia.org/wiki/Cat_righting_reflex

2
The curl of both sides of the third equation gives

Vector calculus tells us that the left-hand side of this equation simplifies to

∇ × ∇ ×
E
= ∇ (∇ ·
E
) − ∇
²
E
= − ∇
²
E

where we also use the first equation. Here ∇
²
is the Laplace operator. Using the fourth equation, the right-hand side becomes

Cancelling out two minus signs and multiplying by
c
²
yields the wave equation for
E
:

A similar calculation yields the wave equation for
H.

1
Specifically,

where
S
_A
and
S
_B
are the entropies in states
A
and
B.

2
The second law of thermodynamics is technically an
inequality
, not an equation. I've included the second law in this book because its central position in science demanded its inclusion. It is undeniably a mathematical
formula
, a loose interpretation of ‘equation' that is widespread outside the technical scientific literature. The formula alluded to in Note 1 of this chapter, using an integral, is a genuine equation. It defines the change in entropy, but the second law tells us what its most important feature is.

3
Brown was anticipated by the Dutch physiologist Jan Ingenhousz, who saw the same phenomenon in coal dust floating on the surface of alcohol, but he didn't propose any theory to explain what he had seen.

1
In the Gran Sasso National Laboratory, in Italy, is a 1300-tonne particle detector called OPERA (oscillation project with emulsion-tracking apparatus). Over two years it tracked 16,000 neutrinos produced at CERN, the European particle physics laboratory in Geneva. Neutrinos are electrically neutral
subatomic particles with a very small mass, and they can pass through ordinary matter with ease. The results were baffling: on average the neutrinos completed the 730-kilometre trip 60 nanoseconds (billionths of a second) sooner than they would have done if they had been travelling at the speed of light. The measurements are accurate to within 10 nanoseconds, but there remains the possibility of some systematic error in the way the times are calculated and interpreted, which is highly complex.

The results have been posted online: ‘Measurement of the neutrino velocity with the OPERA detector in the CNGS beam' by the OPERA Collaboration,
http://arxiv.org/abs/1109.4897

This article does not claim to have disproved relativity: it merely presents its observations as something that the team cannot explain with conventional physics. A non-technical report can be found at
http://www.nature.com/news/2011/110922/full/news.2011.554.html

A possible source of systematic error, related to differences in the force of gravity at the two laboratories, is proposed at

http://www.nature.com/news/2011/111005/full/news.2011.575.html
but the OPERA team disputes this suggestion.

Most physicists think that, despite the great care exercised by the researchers, some systematic error is involved. In particular, previous observations of neutrinos from a supernova seem to conflict with the new ones. The resolution of the controversy will require independent experiments, and these will take several years. Theoretical physicists are already analysing potential explanations ranging from minor, well-known extensions of the standard model of particle physics to exotic new physics in which the universe has more dimensions than the usual four. By the time you read this, the story will already have moved on.

2
A thorough explanation is given by Terence Tao on his website:
http://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/

The derivation of the equation involves five steps:

(a) Describe how space and time coordinates transform when the frame of reference is changed.

(b) Use this description to work out how the frequency of a photon transforms when the frame of reference is changed.

(c) Use Planck's law to work out how the energy and momentum of a photon transform.

(d) Apply conservation of energy and momentum to work out how the energy and momentum of a moving body transform.

(e) Fix the value of an otherwise arbitrary constant in the calculation by comparing the results with Newtonian physics when the velocity of the body is small.

3
Ian Stewart and Jack Cohen.
Figments of Reality
, Cambridge University Press, Cambridge 1997, page 37.

4
http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

5
A few didn't see it that way. Henry Courten, reanalysing photographs of the 1970 solar eclipse, reported the existence of at least seven very tiny bodies in
close orbits round the Sun – perhaps evidence of a thinly populated inner asteroid belt. No conclusive evidence of their existence has been found, and they would have to be less than 60 kilometres across. The objects seen in the photographs may just have been passing small comets or asteroids in eccentric orbits. Whatever they were, they weren't Vulcan.

6
The vacuum energy in a cubic centimetre of free space is estimated to be 10
^−15
joules. According to quantum electrodynamics it should in theory be 10
¹⁰⁷
joules – wrong by a factor of 10
¹²²
.