Special Relativity “Meets”
the Uncertainty Principle
Pavle I. Premović
Laboratory for Geochemistry,
Cosmochemistry&Astrochemistry,
University of Niš, pavleipremovic@yahoo.com, Niš, Serbia
Heisenberg’s
uncertainty principle is one of the fundamental concepts of quantum mechanics
(QM). This principle defines a
relationship between the uncertainties of energy (ΔE) and time (Δt) and it is given in expression form,
ΔEΔt ≥ ћ
where ћ (=
h/2π
= 1.05 × 10-34 J sec) is the reduced
Planck’s constant and h is the
Planck’s constant (6.63 × 10-34 J
sec). This expression is valid for moving objects only.[1]
Besides ћ, the energy-time uncertainty expression includes h (the absolute
maximum), h/2, ћ and ћ/2 (the absolute minimum), depending on the case.
The above expression can be written as follows
ΔEΔt = (1J × 1sec)QM ≥ ћ
or
(1J × 1sec)QM ≥ ћ
… (1).
where the lower script denotes that (1J ×
1sec) is a QM term.
If the uncertainty of the measured energy of 1J increases, the uncertainty for the measured time
interval of 1 sec decreases. Conversely, if the uncertainty of the measured energy
of 1J decreases, the uncertainty in the time interval of 1 sec increases. In
other words, (1J × 1sec)QM is always larger or equal to Planck’s constant
ћ. Therefore, equation (1) represents a new mathematical form of the
energy-time uncertainty.
Time dilation and length contraction are two important effects of
Special relativity (SR). These two effects depend upon the second postulate of
this theory that the speed of light c
(= 2.99792 × 108 m sec-1)
is the same in all inertial frames of
reference. From this postulate, time dilation, as well as length contraction, inevitably results [1].
Simply speaking time
dilation can be defined as a relativistic phenomenon in that a clock at rest shows
the time interval T0 which
is shorter with respect to the time interval Tυ when it is moving with a relative
speed υ. In equation
Tυ
= T0 /√(1 – υ2/c2) … (2)
where 1/√1- υ2/c2) is the Lorentz-Einstein or
the time dilation factor γ.
As we noted above, another important
relativistic phenomenon is length contraction. The formula for determining this
contraction is
Lυ = L0√(1 – υ2/c2) … (3)
where
L0
is the length of the moving object in its rest frame of reference, Lυ is the length of this object measured
by us in our rest frame of reference and υ is the relative velocity between these two frames
of reference.
Multiplying the equations (2) and (3), we
have
Lυ × Tυ = L0 × T0 = constant.
Hence
(1m)0 × (1sec)0
= (1m)υ × (1sec)υ (= constant)
where
(1m)0 and (1sec)0 represent the length of a moving
object of 1m measured in its frame of reference and a time interval of 1 sec measured
also in this frame. On the right side of this expression, (1m)υ and (1sec)υ
represent the length of this object and the time interval of its frame but measured
by us in our rest frame. The relative velocity between these two frames of
reference is υ.
This
can be written as follows
(1m × 1sec)SR = α … (4).
where
α is a constant and the lower script denotes that (1m × 1sec) is an SR term. Of
course, this constant has the same value 1m sec for all observers, or in
different frames of reference, in other words, it is invariant.
According to eqn. (4), if the length of the object
of 1m increases with a speed υ,
the time interval of 1 sec decreases and vice versa, so that (1m × 1sec)SR
is always the same or is equal to the constant α = (1m)0
× (1sec)0 = 1m sec. Therefore,
this can be interpreted as the length-time uncertainty of the Special theory of
relativity.
The relativistic formula for the variation of mass with velocity
υ is
as follows
mυ = m0/√(1 – υ2/c2) … (5)
where mυ
is the mass of a moving object and m0 is its rest mass. Multiplying
this equation and eqn. (3) we obtain
mυLυ = m0L0 =
constant.
Hence,
(1m)0 ×
(1kg)0 = (1m)υ × (1kg)υ
(= constant)
where
(1m)0 and (1kg)0 represent a length of 1m and a
mass of 1 kg of a moving object measured in its frame of reference. On the
right side of this expression, (1m)υ
and (1kg)υ represent the length
and the mass of this object but measured by us in our rest frame. The relative
velocity between these two frames of reference is υ.
In
equation form,
(1kg × 1m)SR
= β … (6)
where
β is also an invariant constant.
According to eqn. (6), if the mass of an object
of 1kg increases with a speed υ, its
length of 1m decreases and vice versa, so that (1kg × 1m)SR is
always the same or is equal to the constant β = (1m)0 × (1kg)0
= 1 kg m. Therefore, eqn. (6) can be interpreted as the mass-length
uncertainty of the Special theory of relativity.
If we multiply eqn. (5) with c2 we
get the following equation for the total relativistic energy of a moving object
Eυ
Eυ = mυc2
= m0c2/√(1 – υ2/c2).
We know that the rest
energy of this object E0 = m0c2, so we write
the above equation as follows
Eυ = E0/√(1 – υ2/c2).
Using a similar approach as above we find that
(1J × 1m)SR = ί … (7)
where
ί (= βc2) is an invariant constant, of course.
According to eqn. (7), if the energy of an object
of 1J increases with a speed υ, its
length of 1m decreases and vice versa, so that (1J × 1m)SR is always
the same or is equal to the constant ί = 1J m. Therefore, eqn. (7) can be
interpreted as the energy-length uncertainty of the Special theory of relativity.
This is how we see that Einstein’s Special
theory of relativity “meets” Heisenberg’s uncertainty principle.
Reference
{1} Taylor E. F. and Wheeler J. A. Spacetime Physics: Introduction to Special Relativity, 2nd ed. Freeman & Company, 1992.
[1] The uncertainty principle applies to all objects but is only significant
for the atomic or subatomic particles.
.jpg)
No comments:
Post a Comment