The Principle of Relativity

Name: The Principle of Relativity
Author: Einstein, Albert

Einstein, Albert

Relativity (Physics)

If we calculate the gravitation-field to a greater order of
approximation and with it the corresponding path of a material particle
of a relatively small (infinitesimal) mass we get a deviation of the
following kind from the Kepler-Newtonian Laws of Planetary motion. The
Ellipse of Planetary motion suffers a slow rotation in the direction of
motion, of amount

            (75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.

In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity
of light, measured in the usual way, _e_, the eccentricity, τ, the time
of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of
amount 43″ per century, corresponding sufficiently to what has been
found by astronomers (Leverrier). They found a residual perihelion
motion of this planet of the given magnitude which can not be explained
by the perturbation of the other planets.




                                 NOTES


                                Note 1.


The fundamental electro-magnetic equations of Maxwell for stationary
media are:—

                     curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)

                     curl E = -1/_c_ ∂B/∂_t_ (2)

                     div D = ρ
                     B = μH
                     div B = 0
                     D = kE

According to Hertz and Heaviside, these require modification in the case
of moving bodies.

Now it is known that due to motion alone there is a change in a vector
_R_ given by

            (∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]

where _u_ is the vector velocity of the moving body and [R_u_] the
vector product of R and _u_.

Hence equations (1) and (2) become

      _c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)

and

        -_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)

which gives finally, for ρ = 0 and div B = 0,

       ∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)

       ∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)

Let us consider a beam travelling along the _x_-axis, with apparent
velocity _v_ (_i.e._, velocity with respect to the fixed ether) in
medium moving with velocity _u__{_x_} = _u_ in the same direction.

Then if the electric and magnetic vectors are proportional to
_e_^{_i_A(_x_ - _vt_)}, we have

    ∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} =
       _u__{_z_} = 0

    Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)

    and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_  (2·21)

Since D = KE and B = μH, we have

          _i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)

          _i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)

          or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)

          μ(_v_ - _u_)H_{_z_} = _c_E_{_y_}  (2·23)

Multiplying (1·23) by (2·23)

                           μK(_v_ - _u_)² = _c²_

Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²

                            ∴ _v_ = _v₀_ + _u_,

Public-domain text, read in full here on John Shaqi.

Opening the text

The Principle of RelativityEinstein, Albert

Science

The Principle of Relativity

Einstein, Albert

Relativity (Physics)

If we calculate the gravitation-field to a greater order of
approximation and with it the corresponding path of a material particle
of a relatively small (infinitesimal) mass we get a deviation of the
following kind from the Kepler-Newtonian Laws of Planetary motion. The
Ellipse of Planetary motion suffers a slow rotation in the direction of
motion, of amount

            (75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.

In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity
of light, measured in the usual way, _e_, the eccentricity, τ, the time
of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of
amount 43″ per century, corresponding sufficiently to what has been
found by astronomers (Leverrier). They found a residual perihelion
motion of this planet of the given magnitude which can not be explained
by the perturbation of the other planets.




                                 NOTES


                                Note 1.


The fundamental electro-magnetic equations of Maxwell for stationary
media are:—

                     curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)

                     curl E = -1/_c_ ∂B/∂_t_ (2)

                     div D = ρ
                     B = μH
                     div B = 0
                     D = kE

According to Hertz and Heaviside, these require modification in the case
of moving bodies.

Now it is known that due to motion alone there is a change in a vector
_R_ given by

            (∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]

where _u_ is the vector velocity of the moving body and [R_u_] the
vector product of R and _u_.

Hence equations (1) and (2) become

      _c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)

and

        -_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)

which gives finally, for ρ = 0 and div B = 0,

       ∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)

       ∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)

Let us consider a beam travelling along the _x_-axis, with apparent
velocity _v_ (_i.e._, velocity with respect to the fixed ether) in
medium moving with velocity _u__{_x_} = _u_ in the same direction.

Then if the electric and magnetic vectors are proportional to
_e_^{_i_A(_x_ - _vt_)}, we have

    ∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} =
       _u__{_z_} = 0

    Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)

    and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_  (2·21)

Since D = KE and B = μH, we have

          _i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)

          _i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)

          or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)

          μ(_v_ - _u_)H_{_z_} = _c_E_{_y_}  (2·23)

Multiplying (1·23) by (2·23)

                           μK(_v_ - _u_)² = _c²_

Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²

                            ∴ _v_ = _v₀_ + _u_,

Public-domain text, read in full here on John Shaqi.

The Principle of Relativity

The Principle of Relativity

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