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R. Barrett and P. P. Delsanto
been known that energy can be converted from one form to another (e.g.
potential energy to kinetic energy to heat ), but never created or destroyed.
Since Einstein, these two quantities, mass and energy, can be considered as
different forms of a single entity. In nuclear reactions, mass can be converted
to energy, and vice versa. The same is also true in chemical reactions, but
the energies involved are so small that any change in mass of the reactants is
unobservable.
Einstein’s formula is remarkable from a philosophical point of view because
it unexpectedly relates together three variables which nobody had ever before
dared to assume were associated, i.e. mass, energy and the velocity of light.
It may be interesting to some readers to recall that a similar kind of unexpected association between apparently unconnected quantities can be found
in mathematics. We refer to Euler’s formula, which establishes a fundamental
relationship between the hitherto unrelated fields of trigonometric functions,
exponentials and complex numbers. 7 The physicist, Richard Feynman, called
Euler’s equation “our jewel ” and “the most remarkable formula in mathematics". It may be a subject of philosophical speculation to investigate the
underlying reasons for these (and a few other) unexpected connections.
A formal demonstration of the mass/energy relationship is beyond the
scope of our book, since it involves a lengthy mathematical treatment. (A
somewhat simpler justification of it can be found in [6].) What is more
instructive for our purposes here is to examine some of its consequences.
We begin by considering the problem of an accelerating rocket ship. In
classical physics, as the rocket consumes fuel and ejects the combustion products, it receives a propulsive force which drives it forwards. As long as this
force remains, the rocket will continue increasing its velocity, and thereby its
kinetic energy. According to Einstein, however, some of this kinetic energy
will be transformed into mass, thus making the rocket ship more difficult to
accelerate further. The closer the rocket’s velocity gets to the velocity of light,
the more pronounced this effect becomes. If the rocket’s speed were ever to
reach c, its mass would be infinite. This is mathematics’ way of telling us that
it is not possible ever to accelerate a rocket, or any other material object, to
speeds greater than or equal to c. The velocity of light is a universal limiting
speed that nothing can exceed.
One may raise the objection that photons are particles, and surely, since
they are particles of light, they travel at speed c. Yes, they do, but they are a
special type of particle, possessing no rest mass. The rest mass of a particle is, as
the name implies, the mass of the particle when it is stationary. When it is in
7 Those of a mathematical bent may check this out in any elementary mathematics textbook, where
they will find: e ix = cos x + i sin x.
R. Barrett and P. P. Delsanto
been known that energy can be converted from one form to another (e.g.
potential energy to kinetic energy to heat ), but never created or destroyed.
Since Einstein, these two quantities, mass and energy, can be considered as
different forms of a single entity. In nuclear reactions, mass can be converted
to energy, and vice versa. The same is also true in chemical reactions, but
the energies involved are so small that any change in mass of the reactants is
unobservable.
Einstein’s formula is remarkable from a philosophical point of view because
it unexpectedly relates together three variables which nobody had ever before
dared to assume were associated, i.e. mass, energy and the velocity of light.
It may be interesting to some readers to recall that a similar kind of unexpected association between apparently unconnected quantities can be found
in mathematics. We refer to Euler’s formula, which establishes a fundamental
relationship between the hitherto unrelated fields of trigonometric functions,
exponentials and complex numbers. 7 The physicist, Richard Feynman, called
Euler’s equation “our jewel ” and “the most remarkable formula in mathematics". It may be a subject of philosophical speculation to investigate the
underlying reasons for these (and a few other) unexpected connections.
A formal demonstration of the mass/energy relationship is beyond the
scope of our book, since it involves a lengthy mathematical treatment. (A
somewhat simpler justification of it can be found in [6].) What is more
instructive for our purposes here is to examine some of its consequences.
We begin by considering the problem of an accelerating rocket ship. In
classical physics, as the rocket consumes fuel and ejects the combustion products, it receives a propulsive force which drives it forwards. As long as this
force remains, the rocket will continue increasing its velocity, and thereby its
kinetic energy. According to Einstein, however, some of this kinetic energy
will be transformed into mass, thus making the rocket ship more difficult to
accelerate further. The closer the rocket’s velocity gets to the velocity of light,
the more pronounced this effect becomes. If the rocket’s speed were ever to
reach c, its mass would be infinite. This is mathematics’ way of telling us that
it is not possible ever to accelerate a rocket, or any other material object, to
speeds greater than or equal to c. The velocity of light is a universal limiting
speed that nothing can exceed.
One may raise the objection that photons are particles, and surely, since
they are particles of light, they travel at speed c. Yes, they do, but they are a
special type of particle, possessing no rest mass. The rest mass of a particle is, as
the name implies, the mass of the particle when it is stationary. When it is in
7 Those of a mathematical bent may check this out in any elementary mathematics textbook, where
they will find: e ix = cos x + i sin x.