Principle of
The Principle of Relativity
作者：Einstein, Albert, 1879-1955; Minkowski, H. (Hermann), 1864-1909; Mahalanobis, P. C. (Prasanta Chandra), 1893-1972 [Author of introduction, etc.]; Bose, Satyendranath, 1894-1974 [Translator]; Saha, Meghnad, 1893-1956 [Translator]
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章节目录
共 203 题- The Principle of Relativity
- THE PRINCIPLE OF RELATIVITY
- ORIGINAL PAPERS BY
- A. EINSTEIN AND H. MINKOWSKI
- TRANSLATED INTO ENGLISH BY
- M. N. SAHA AND S. N. BOSE
- WITH A HISTORICAL INTRODUCTION BY
- PRINTED BY ATULCHANDRA BHATTACHARYYA,
- AT THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA
- TABLE OF CONTENTS
- 1. Historical Introduction i-xxiii
- 2. On the Electrodynamics of Moving Bodies 1-34
- 3. Albrecht Einstein 35-39
- 4. Principle of Relativity 1-52
- 5. Appendix to the above by H. Minkowski 53-88
- 6. The Generalised Principle of Relativity 89-163
- 7. Notes 165-185
- HISTORICAL INTRODUCTION
- Conclusion
- P. C. MAHALANOBIS.
- INTRODUCTION.
- I.—KINEMATICAL PORTION.
- In accordance with experience we shall assume that the magnitude
- Substituting these values of _t_ in the equation for ξ, we obtain
- Therefore
- If for _x′_, we substitute its value _x_ - _tv_, we obtain
- II.—ELECTRODYNAMICAL PART.
- X′ = X₀ sin Φ′
- L′ = L₀ sin Φ′
- If Φ = 0, this reduces to the simple form
- If Φ = 0, we have the simple formula:—
- A″ = A′, cos Φ″ = -cos Φ″, ν″ = ν′
- As a first approximation, we obtain
- Let us start from the equations:—
- Principle of Relativity
- INTRODUCTION.
- NOTATIONS.
- Further I shall write
- Therefore
- Putting
- The solution of the equations (10), (11), (12) leads to
- Now it is quite apparent that the system of values
- Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_
- Then the fundamental Equations can be written as
- The first Axion shall be,—
- Shortly I can signify the third axiom as:—
- I) curl _m_ - ∂_e_/∂_t_ = C₁,
- II) div _e_ = ρ
- III) curl E + ∂M/∂_t_ = 0
- IV) div M = 0
- A system of magnitudes _a__{_h_ _k_} formed into the matrix
- 2⁰ If we have two matrices
- For the transposed matrix of C = BA, we have Ċ = ḂĀ
- A matrix
- Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_
- 4⁰. A linear transformation
- By the transformation A, the expression
- From the condition (39) we obtain
- A⁻¹ = Ā,
- Then it follows immediately that the system of six magnitudes (44)
- In Φ = ωF, we have a space time vector of the first kind with components
- Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄
- Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄
- Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄
- Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃
- Among these there is the relation
- Let us now take the vector of the 1st kind
- Then by applying rule (45), we have
- The vector Ω fulfils the relation
- I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.
- For this matrix I shall use the shortened from “lor.”[25]
- Therefore it follows that
- If _s_ is a space-time vector of the 1st kind, then
- In case of a Lorentz transformation A, we have
- I shall now collect the results.
- Let ω denote the space-time vector of the first kind
- F the space-time vector of the second kind (M,-_i_E)
- Then we shall have
- In order to express in a real form, we write
- Here we have
- The expression on the right-hand side of (81), which equals
- Then is
- Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |
- T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃
- X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃
- According to 71), we have
- The space-time vector of the first kind
- According to 78), S = L + _f_F, and it follows that
- The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2)
- To the increment _dt_ of the time of P corresponds the increment
- If the _sichel_ reduces to a point, then the differential equation
- Further let us form the integral
- N = ∫ ∫∫∫ ν _dx dy dz dt_ (7)
- Then the integral (7) can be denoted by
- If we now introduce the method of writing with indices, we shall have
- We shall now subject the value of the differential quotient
- Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13)
- The sum which comes forth here, written in real magnitudes, is
- X, Y, Z, ·_i_T
- Now neglecting _c⁻⁴_ with regard to 1, it follows that
- SPACE AND TIME
- II
- 1:(√(1 - v²/_c²_)) in length, in the direction of its motion
- If (_dc_/_dt_) = _v_, an easy calculation gives that
- OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))
- P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP
- III
- IV
- If we substitute the expression in (1) we get
- 5. Contra-variant and co-variant Four-vector.
- A_{β}^{δ} = A_{αβ}^{αδ} = (∑_{α} A_{αβ}^{αδ})
- Proof:—According to the above assumption, for any substitution we have
- From the inversion of (9) we have however
- Substitution of this for B^{μν} in the above equation gives
- According to the well-known law of Determinants
- If we introduce it in our expression, we get
- According to the law of multiplication of determinants, we have
- On the other hand we have
- From this by applying the law of multiplication twice, we obtain
- On the other hand the law of transformation of the volume element
- In place of (18) it follows then simply that
- Example:—
- A^{μ} = _g_^{μσ} A_{σ}
- A = _g__{μν} A^{μν}
- A^{μν} = _g_^{μα} _g_^{νβ} A_{αβ}
- A_{μν} = _g__{μα} _g__{νβ} A^{αβ}
- Similarly
- Its equation is
- Then (20) can be replaced by
- But
- So we get by the substitution of δω in (20a), remembering that
- Since
- A_{μ} = ∂φ/∂_x__{μ}
- Substituting the value of ψ, we get
- Now the right hand side of (25) multiplied by ψ is
- Through addition follows the tensor character of
- The last form follows from the first when we remember that
- From (28), it follows that
- By substituting the right-hand side of (34) in (29), we get
- According to (31) and (29), the last member can take the form
- On the account of the symmetry of
- If further A^{ρσ} or is symmetrical it is reduced to
- C. THE THEORY OF THE GRAVITATION-FIELD
- H = _g_^{μν} γ^{α}_{μβ} γ^{β}_{να}
- We have at first
- But
- Therefore
- If (47b) is multiplied by _g__{σ}^{μν}, since
- Therefore in the place of the equations (47), we obtain
- We thus get instead of (51), the tensor-equation
- So that remembering (54) we have
- From (55) and (52a) it follows that
- D. THE “MATERIAL” PHENOMENA.
- T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
- From (59), it follows that the system of equations
- We introduce the contra-variant six-vector F_{αβ} by the equation
- If we put
- J^1 = _i__{_x_} ... J^4 = ρ
- Let us form the inner-product
- K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
- F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
- Owing to symmetry, this expression can also be written in the form
- The first of these terms can be written shortly as
- If we take all the three terms together, we get the relation
- So that remembering (54) we have
- From (55) and (52a) it follows that
- D. THE “MATERIAL” PHENOMENA.
- T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
- From (59), it follows that the system of equations
- We introduce the contra-variant six-vector F_{αβ} by the equation
- If we put
- J^1 = _i__{_x_} ... J^4 = ρ
- Let us form the inner-product
- K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
- F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
- Owing to symmetry, this expression can also be written in the form
- The first of these terms can be written shortly as
- If we take all the three terms together, we get the relation
- The last of the equations (53) thus leads to
- On account of (68a) we have
- From both these relations it follows as a first approximation that
- 1 = _g₄₄__dx₄²_
- B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_
- The calculation gives
- NOTES
- Hence equations (1) and (2) become
- Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)
- Since D = KE and B = μH, we have
- Multiplying (1·23) by (2·23)
- Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²
- Equations (_i_) and (_ii_) become when expanded into Cartesians:—
- Now substitute
- In a corresponding way we have
- We have
- We can verify that
- Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get
- It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]
- Now remembering that generally
- Leaving aside the first column we obtain
- Now since ρ₀ = ρ√(1 - V²/_c²_)
- We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]
- We have therefore
- Generally we have