OFFSET
9,1
COMMENTS
Also decimal expansion of the speed b = c/sqrt(2) in SI units (meter/second), where c = 299792458 (m/s) is the speed of light in vacuum (A003678).
A particle (or object) with speed b has the property that its relativistic momentum equals the momentum of a virtual photon whose energy equals the rest energy of the particle. Also its relativistic de Broglie wavelength equals the Compton wavelength for the particle and therefore equals the wavelength of the photon mentioned above.
More generally it appears that the speed b is a critical speed for several relativistic magnitudes of the particle. Explanation: consider a table of relativistic magnitudes in which every formula is written as the product of a dimensionless factor and a constant with the same dimensions as the relativistic magnitude. For instance, for the relativistic momentum we write the formula p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c instead of the standard formula p = [1/(1 - v^2/c^2)^(1/2)]*m_0*v. See below:
Table 1.
----------------------------------------------------
Relativistic
magnitude Formula
----------------------------------------------------
Speed.........: v = [v/c]*c
Group velocity: g = [v/c]*c
Length........: L = [1/γ]*L_0
Momentum......: p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c
Wavenumber....: k = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c/h
Wavelength....: W = [(c^2/v^2 - 1)^(1/2)]*h/(m_0*c)
Time interval.: t = γ*t_0
Mass..........: m = γ*m_0
Energy........: E = γ*m_0*c^2
Frequency.....: f = γ*m_0*c^2/h
Phase velocity: w = [c/v]*c
Kinetic energy: K = [γ - 1]*m_0*c^2
----------------------------------------------------
Where:
v is the speed of the object or particle.
c is the speed of light in vacuum (A003678).
h is the Planck constant (A003676).
L_0 is the length at rest of the object or the length at rest of a virtual cube which contains the particle.
t_0 is the time interval at rest.
W is the relativistic de Broglie wavelength assuming that W = h/p.
γ = [1/(1 - v^2/c^2)^(1/2)] is the Lorentz factor.
Then table 1 can be unified as shown below:
Table 2. Table 3.
------------------------------------ -----------------
Relativistic
magnitude Formula Formula
------------------------------------ -----------------
Speed.........: v = sin(x) * c v = sin(x) * v’
Group velocity: g = sin(x) * c g = sin(x) * g’
Length........: L = cos(x) * L_0 L = cos(x) * L’
Momentum......: p = tan(x) * m_0*c p = tan(x) * p’
Wavenumber....: k = tan(x) * 1/W_C k = tan(x) * k’
Wavelength....: W = cot(x) * W_C W = cot(x) * W’
Time interval.: t = sec(x) * t_0 t = sec(x) * t’
Mass..........: m = sec(x) * m_0 m = sec(x) * m’
Energy........: E = sec(x) * E_0 E = sec(x) * E’
Frequency.....: f = sec(x) * E_0/h f = sec(x) * f’
Phase velocity: w = csc(x) * c w = csc(x) * w’
Kinetic energy: K = ese(x) * E_0 K = ese(x) * K’
------------------------------------ -----------------
Where:
W_C = h/(m_0*c) is the Compton wavelength for the particle (for the electron see A230436, for the proton see A230845).
ese(x) = sec(x) - 1.
Table 2 is simpler than table 1 because the relativistic factors are written as trigonometric functions of the angle x assuming that sin(x) = v/c and that 0 < x < Pi/2.
Table 3 lists the simplest formulas in which the values of the constants have been interpreted as the values of the magnitudes of a virtual photon whose energy E' = h*f' is equivalent to E_0 = m_0*c^2, the rest energy of the particle.
A visualization of the relationship between the relativistic magnitudes, the quantum constants and the trigonometric functions is obtained using the first quadrant of the trigonometric circle according to the simplest table, see below:
Table 4.
-----------------------------------
sin(x) = v/v' = g/g'
cos(x) = L/L'
tan(x) = p/p' = k/k'
cot(x) = W/W'
sec(x) = t/t' = m/m' = E/E' = f/f'
csc(x) = w/w'
ese(x) = K/K'
-----------------------------------
Finally we can write that b is a critical speed because:
If v = b, for instance, we have that:
1) v/v’ = L/L’ = sin(Pi/4) = cos(Pi/4) = 2^(1/2)/2.
2) p/p’ = W/W’ = tan(Pi/4) = cot(Pi/4) = 1.
3) E/E’ = w/w’ = sec(Pi/4) = csc(Pi/4) = 2^(1/2).
Otherwise if v < b we have that:
v/v’ < L/L’ and p/p’ < W/W’ and E/E’ < w/w’.
Otherwise if v > b we have that:
v/v’ > L/L’ and p/p’ > W/W’ and E/E’ > w/w’.
LINKS
G. C. Greubel, Table of n, a(n) for n = 9..10008
Omar E. Pol, A trigonometric model of the Relativity: Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8
Omar E. Pol, A table of relativistic factors
Wikipedia, Quantum mechanics
Wikipedia, Special relativity
Wikipedia, Trigonometry
EXAMPLE
211985280.000383... m/s.
MATHEMATICA
RealDigits[149896229*Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 26 2018 *)
PROG
(PARI) 149896229*sqrt(2) \\ G. C. Greubel, Jan 26 2018
(Magma) 149896229*Sqrt(2); // G. C. Greubel, Jan 26 2018
CROSSREFS
KEYWORD
AUTHOR
Omar E. Pol, Nov 10 2013
STATUS
approved