# Chapter 2: Special Theory of Relativity¶

## Example 2.2, Page 34¶

In :
import math

#Variable declaration
ly = 9.46e+015;    # Distance travelled by light in an year, m
c = 3e+008;    # Speed of light, m/s
L = 4.30*ly;    # Distance of Alpha Centauri from earth, m
T0 = 16*365.25*24*60*60;    # Proper time in system K_prime, s

#Calculations
# As time measured on earth, T = 2*L/v = T0_prime/sqrt(1-(v/c)^2), solving for v
v = math.sqrt(4*L**2/(T0**2+4*L**2/c**2));    # Speed of the aircraft, m/s
gama = 1/math.sqrt(1-(v/c)**2);    # Relativistic factor
T = gama*T0/(365.25*24*60*60);    # Time interval as measured on Earth, y

#Results
print "The speed of the aircraft = %4.2e m/s" %v
print "The time interval as measured on earth = %4.1f y"%T

The speed of the aircraft = 1.42e+08 m/s
The time interval as measured on earth = 18.2 y


## Example 2.3, Page 38¶

In :
import math

#Variable declaration
L0 = 4.30;    # Distance of Alpha Centauri from earth, ly
c = 3e+008;    # Speed of light, m/s
T = 8;    # Proper time in system K_prime, y

#Calculations
# As v/c = L0*sqrt(1-(v/c)^2)/(c*T) or bita = L0*sqrt(1-bita^2)/(c*T), solving for bita
bita = math.sqrt(L0**2/(T**2 + L0**2));    # Boost parameter
v = L0*math.sqrt(1-bita**2)/T;    # Speed of the aircraft, c units

#Results
print "The boost parameter = %5.3f"%bita
print "The speed of the aircraft = %5.3fc units"%v

The boost parameter = 0.473
The speed of the aircraft = 0.473c units


## Example 2.4, Page 40¶

In :
import math

#Variable declaration
c = 1;    # For simplicity assume speed of light to be unity, m/s
bita = 0.600;    # Boost parameter
gama = 1/math.sqrt(1-bita**2);    # Relativistic factor
u_x_prime = 0;    # Speed of the protons in spaceship frame along x-axis, m/s
u_y_prime = 0.99*c;    # Speed of the protons in spaceship frame along y-axis, m/s
u_z_prime = 0;    # Speed of the protons in spaceship frame along z-axis, m/s
v = 0.60*c;    # Speed of the spaceship w.r.t. space station, m/s

#Calculations
u_x = (u_x_prime + v)/(1 + v/c**2*u_x_prime);    # Speed of protons in space station frame along x-axis, m/s
u_y = u_y_prime/(gama*(1 + v/c**2*u_x_prime));    # Speed of protons in space station frame along y-axis, m/s
u_z = u_z_prime/(gama*(1 + v/c**2*u_x_prime));    # Speed of protons in space station frame along y-axis, m/s
u = math.sqrt(u_x**2 + u_y**2 + u_z**2);    # The speed of the protons measured by an observer in the space station, m/s

#Result
print "The speed of the protons measured by an observer in the space station = %5.3fc units"%u

The speed of the protons measured by an observer in the space station = 0.994c units


## Example 2.5, Page 45¶

In :
#Variable declaration
c = 2.998e+008;    # Speed of light in free space, m/s
v = 7712;    # Speed of the space shuttle, m/s
bita = v/c;    # Boost parameter
T_loss = 295.02;    # Total measured loss in time, ps/sec
Add_T_loss = 35.0;    # Additional time loss for moving clock from general relativity prediction, ps/s

#Calculations
# From time dilation relation, T0_prime = T*sqrt(1 - bita^2), solving for (T - T0_prime)/T
Calc_T_loss = bita**2/2*1e+012;    # Expected time lost per sec by the moving clock, ps/sec
Measured_T_loss = T_loss + Add_T_loss;    # Total measured loss in time per sec as per special relativity, ps/s
percent_T_loss = (Calc_T_loss - Measured_T_loss)/Calc_T_loss*100;    # Percentage deviation of measured and the calculated time loss per sec
T = 6.05e+05;    # Total time of the seven-day mission, s
delta_T = Calc_T_loss*1e-012*T;    # The total time difference between the moving and stationary clocks during the entire shuttle flight, s

#Results
print "The expected time lost per second for the moving clock = %6.2f ps"%Calc_T_loss
print "The percentage deviation of measured and the calculated time loss per sec for moving clock = %3.1f percent"%percent_T_loss   #answer differs due to rounding errors
print "The total time difference between the moving and stationary clocks during the entire shuttle flight = %3.1f ms"%(delta_T/1e-003)

The expected time lost per second for the moving clock = 330.86 ps
The percentage deviation of measured and the calculated time loss per sec for moving clock = 0.3 percent
The total time difference between the moving and stationary clocks during the entire shuttle flight = 0.2 ms


## Example 2.8, Page 57¶

In :
import math

#Variable declaration
f0 = 1;    # For simplicity assume frequency of the signals sent by Frank, Hz
# Outbound trip
bita = -0.8;    # Boost parameter for receding frames

#Calculations&Results
f = math.sqrt(1+bita)/math.sqrt(1-bita)*f0;    # The frequency of the signals received by Mary in outbound trip, Hz
print "The frequency of the signals received by Mary in outbound trip = f0/%d", math.ceil(f*9)
# Return trip
bita = +0.8;    # Boost parameter for approaching frames
f = math.sqrt(1+bita)/math.sqrt(1-bita)*f0;    # The frequency of the signals received by Mary in return trip, Hz
print "The frequency of the signals received by Mary in return trip = %df0"%f

 The frequency of the signals received by Mary in outbound trip = f0/%d 3.0
The frequency of the signals received by Mary in return trip = 3f0


## Example 2.11, Page 64¶

In :
import math

#Variable declaration
q = 1.6e-019;    # Charge on an electron, C
V = 25e+003;    # Accelerating potential, volt
K = q*V;    # Kinetic energy of electrons, J
m = 9.11e-031;    # Rest mass of an electron, kg
c = 3.00e+08;    # Speed of light, m/s

#Calculations
# From relativistic kinetic energy formula, K = (gama - 1)*m*C^2, solving for gama
gama = 1 + K/(m*c**2);    # Relativistic factor
bita = math.sqrt((gama**2-1)/gama**2);    # Boost parameter
u = bita*c;    # Speed of the electrons, m/s
# From non-relativistic expression, K = 1/2*m*u^2, solving for u
u_classical = math.sqrt(2*K/m);    # Non-relativistic speed of the electrons, m/s
u_error = (u_classical - u)/u_classical*100;    # Percentage error in speed of electrons, m/s

#Results
print "The relativistic speed of the accelerated electrons = %4.2e m/s"%u
print "The classical speed is about %d percent greater than the relativistic speed"%math.ceil(u_error)

The relativistic speed of the accelerated electrons = 9.04e+07 m/s
The classical speed is about 4 percent greater than the relativistic speed


## Example 2.13, Page 69¶

In :
import math

#Variable declaration
c = 1;    # For simplicity assume peed of light to be unity, m/s
K = 2.00;    # Kinetic energy of protons, GeV
E0 = 0.938;    # Rest mass of a proton, GeV
E = K + E0;    # Total energy of the proton, GeV

#Calculations
# From relativistic mass energy relation, E^2 = (p*c)^2+E0^2, solving for p
p = math.sqrt(E**2 - E0**2)/c;    # Momentum of the protons, GeV/c
# As E = gama*E0, solving for gama
gama = E/E0;    # Relativistic factor
bita = math.sqrt((gama**2-1)/gama**2);    # Boost parameter
v = bita*3.00e+08;    # Speed of 2 GeV proton, m/s

#Results
print "The energy of each initial proton = %5.3f GeV"%E
print "The momentum of each initial proton = %4.2f GeV/c"%p
print "The speed of each initial proton = %3.1e m/s"%v
print "The relativistic factor, gama = %4.2f"%gama
print "The boost parameter, beta = %5.3f"%bita

The energy of each initial proton = 2.938 GeV
The momentum of each initial proton = 2.78 GeV/c
The speed of each initial proton = 2.8e+08 m/s
The relativistic factor, gama = 3.13
The boost parameter, beta = 0.948


## Example 2.15, Page 71¶

In :
#Variable declaration
E_d = 1875.6;    # Rest mass energy of the deuterium, MeV
E_pi = 139.6;    # Rest mass energy of the pion, MeV
E_p = 938.3;    # Rest mass energy of the proton, MeV

#Calculation
K = 1./2*(E_d + E_pi - 2*E_p);    # Minimum kinetic energy of the protons, MeV

#Result
print "The minimum kinetic energy of the protons = %2d MeV"%K

The minimum kinetic energy of the protons = 69 MeV


## Example 2.16, Page 72¶

In :
#Variable declaration
u = 931.5;    # Energy equivalent of 1 amu, MeV
c = 1;    # Speed of light in vacuum, m/s

#Calculations
m_e = 0.000549*u;    # Rest mass of an electron, MeV/c^2
m_p = 1.007276*u;    # Rest mass of a proton, MeV/c^2
m_n = 1.008665*u;    # Rest mass of a neutron, MeV/c^2
m_He = 4.002603*u;    # Rest mass of a helium, MeV/c^2
M_He = m_He - 2*m_e;    # Mass of He nucleus, MeV/c^2
E_B_He = (2*m_p + 2*m_n - M_He)*c**2;    # Binding energy of the He nucleus, MeV

#Result
print "The binding energy of the He nucleus = %4.1f MeV"%E_B_He

The binding energy of the He nucleus = 28.3 MeV


## Example 2.17, Page 72¶

In :
#Variable declaration
u = 931.5;    # Energy equivalent of 1 amu, MeV/u
c = 1;    # For simplicity assume speed of light in vacuum to be unity, m/s
E_B = 4.24;    # The dissociationenergy of the NaCl molecule, MeV

#Calculations
M = 58.44*u;    # Energy corresponding to molecular mass of NaCl, MeV
f_r = E_B/M;    # The fractional mass increase of the Na and Cl atoms

#Result
print "The fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl = %4.1e"%(f_r/1e+006)

The fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl = 7.8e-11


## Example 2.18, Page 72¶

In :
import math

#Variable declaration
c = 1;    # For simplicity assume speed of light in vacuum to be unity, m/s
E0_n = 940;    # Rest energy of a neutron, MeV
E0_pi = 140;    # Rest energy of a pion, MeV
p_n = 4702;    # Momentum of the neutron, MeV/c
p_pi = 169;    # Momentum of the pion, MeV/c

#Calculations
E_n = math.sqrt((p_n*c)**2+E0_n**2);    # Total energy of the neutron, MeV
E_pi = math.sqrt((p_pi*c)**2+E0_pi**2);    # Total energy of the pion, MeV
E = E_n + E_pi;    # Total energy of the reaction, MeV
p_sigma = p_n + p_pi;    # Momentum of the sigma particle, MeV/c
E0_sigma = math.sqrt(E**2 - (p_sigma*c)**2);    # Rest mass energy of the sigma particle, MeV
m_sigma = E0_sigma/c**2;    # Rest mass of sigma particle, MeV/c^2;
K = E - E0_sigma;    # Kinetic energy of sigma particle, MeV

#Results
print "The rest mass of sigma particle = %4d MeV/c^2"%math.ceil(m_sigma)
print "The kinetic energy of sigma particle = %4d MeV"%math.ceil(K)

#Answers differ due to rounding errors

The rest mass of sigma particle = 1192 MeV/c^2
The kinetic energy of sigma particle = 3824 MeV