30
3 Mechanical Aspects of Biosystems
are selected for each. All other quantities with physical units can be defined in terms
of the units for each of these three. When there is no compelling reason to select
otherwise, we will be using the ‘cgs’ system of units, with centimeter-gram-second
unit sizes, or, secondarily, the ‘mks’ system, meter-kilogram-second.
For systems moving fast compared to us, such as life forms moving in relativistic
spaceships, we apply the extension of Newton’s laws given by Einstein. In Einstein’s
Special Theory of Relativity, a generalization of Newtonian mechanics to bodies
with relative motions close to the speed of light, the concept of space cannot be
separated from the concept of time. That idea leaves only two independently defined
units, often taken to be length and mass. Time units are converted to length units by
multiplying the time measure by the speed of light, c. The units of energy can be
expressed in mass units, using Einstein’s famous discovery in the analysis of Special
Relativity that E = mc 2 . 4
Since Special Relativity limits the speed of ordinary particles and quanta of
interaction to be no greater than a universal constant c, Einstein knew that Newton’s
Theory of Gravity also had to be corrected, since Newtonian gravity acts instantaneously over long distances. By thinking about rides in falling and accelerated
elevators, he could explain why inertial mass was the same as gravitational mass
with the proposal that the local effects of acceleration are indistinguishable from
gravity. Finally, using his Special Theory to compare the circumference to the
radius of a fixed platform compared to a rotating platform, he concluded that
space-time must be curved in an accelerated frame. In this way, he knew that nonuniform gravity must be associated with space-time curvature. His General Theory
of Relativity is the mathematical embodiment of these propositions.
In Einstein’s General Theory of Relativity, which associates gravity with a
curvature of space-time due to nearby masses, the concept of mass is not independent of length (mass produces gravitationally-curved space, with curvature defined
by the inverse of a length). Moreover, gravitational mass in Einstein’s theory is
identical to inertial mass. 5 Within Einstein’s theory, we can convert all physical
units to a length. For example, using the universal gravitational constant: G =
6.67408 × 10 −11 N m 2 /kg 2 , we can multiply mass by 2 G/c 2 to get a length. If a
mass were to be compressed into a ball of this radius, it would be a black hole. For
the Sun, 2 GM/c
2
= 2.953 km.
So, in the present state of our theories, there remains only one scale with units,
conventionally taken as length units.
4 Einstein was the first to see, in 1905, that mass holds a very large quantity of energy. A chemical
reaction which releases energy causes an almost imperceptible loss of mass. Nuclear reactions can
produce much more energy from mass than chemical reactions because much more binding energy
is involved.
5 Inertial and gravitational mass have been shown equivalent in a number of very precise
measurements. See, e.g., P.G. Roll, R. Krotkov, and R.H. Dicke, The Equivalence of Inertial and
Passive Gravitational Mass, Ann Phys 26, 442–517 (1964).
3 Mechanical Aspects of Biosystems
are selected for each. All other quantities with physical units can be defined in terms
of the units for each of these three. When there is no compelling reason to select
otherwise, we will be using the ‘cgs’ system of units, with centimeter-gram-second
unit sizes, or, secondarily, the ‘mks’ system, meter-kilogram-second.
For systems moving fast compared to us, such as life forms moving in relativistic
spaceships, we apply the extension of Newton’s laws given by Einstein. In Einstein’s
Special Theory of Relativity, a generalization of Newtonian mechanics to bodies
with relative motions close to the speed of light, the concept of space cannot be
separated from the concept of time. That idea leaves only two independently defined
units, often taken to be length and mass. Time units are converted to length units by
multiplying the time measure by the speed of light, c. The units of energy can be
expressed in mass units, using Einstein’s famous discovery in the analysis of Special
Relativity that E = mc 2 . 4
Since Special Relativity limits the speed of ordinary particles and quanta of
interaction to be no greater than a universal constant c, Einstein knew that Newton’s
Theory of Gravity also had to be corrected, since Newtonian gravity acts instantaneously over long distances. By thinking about rides in falling and accelerated
elevators, he could explain why inertial mass was the same as gravitational mass
with the proposal that the local effects of acceleration are indistinguishable from
gravity. Finally, using his Special Theory to compare the circumference to the
radius of a fixed platform compared to a rotating platform, he concluded that
space-time must be curved in an accelerated frame. In this way, he knew that nonuniform gravity must be associated with space-time curvature. His General Theory
of Relativity is the mathematical embodiment of these propositions.
In Einstein’s General Theory of Relativity, which associates gravity with a
curvature of space-time due to nearby masses, the concept of mass is not independent of length (mass produces gravitationally-curved space, with curvature defined
by the inverse of a length). Moreover, gravitational mass in Einstein’s theory is
identical to inertial mass. 5 Within Einstein’s theory, we can convert all physical
units to a length. For example, using the universal gravitational constant: G =
6.67408 × 10 −11 N m 2 /kg 2 , we can multiply mass by 2 G/c 2 to get a length. If a
mass were to be compressed into a ball of this radius, it would be a black hole. For
the Sun, 2 GM/c
2
= 2.953 km.
So, in the present state of our theories, there remains only one scale with units,
conventionally taken as length units.
4 Einstein was the first to see, in 1905, that mass holds a very large quantity of energy. A chemical
reaction which releases energy causes an almost imperceptible loss of mass. Nuclear reactions can
produce much more energy from mass than chemical reactions because much more binding energy
is involved.
5 Inertial and gravitational mass have been shown equivalent in a number of very precise
measurements. See, e.g., P.G. Roll, R. Krotkov, and R.H. Dicke, The Equivalence of Inertial and
Passive Gravitational Mass, Ann Phys 26, 442–517 (1964).