Annex 1

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Annex 1 Lorentz transformations We define two reference inertial systems S and S', such that Si t = 0, t' = 0, x = x' [A1.1] Si x' = 0, v = x/t [A1.2] The condition of special relativity is c = x/t = x'/t'. For it to be fulfilled, whether the relative movement of one system with respect to the other is according to the positive axes of abscissas OX and O'X 'or in the opposite direction, it must be verified: x - ct = x' - ct' x + ct = x' + ct' In general: x' - ct' = Îą (x - ct) x' + ct' = β (x + ct) Adding and subtracting both equations, is obtained: x' = (Îą + β)/2x - (Îą - β)/2ct ct' = -(Îą - β)/2x + (Îą + β)/2ct Doing A = (Îą + β)/2 and B = (Îą - β)/2, the equations remain like this: x' = Ax - Bct = A (x - B/Act) [A1.3] ct' = -Bx + Act = A (-B/Ax + ct) [A1.4] We eliminate the parameter c from particular solutions of both equations. In equation [A1.3], for t = 0, x' = Ax. For the particular case x' = 1, x = 1/A [A1.5] In equation [A1.4], for t' = 0, ct = B/Ax. Substituting this value in the equation [A1.1], we obtain: x' = Ax (1 - B2/A2). For the particular case x = 1, x' = A (1 - B2/A2) [A1.6] Applying the condition x = x' of [A1.1] in equations [A1.5] and [A1.6]: 1/A = A (1 - B2/A2) A = (1 - B2/A2)-1/2 [A1.7] Applying the condition x' = 0 de [A1.2] in the equation [A1.3]: x/t = Bc/A→ v/c = B/A [A1.8] Substituting B/A of [A1.8] in [A1.7], A will be: A = (1 -v2/c2)-1/2 [A1.9] These last two values, B/A of [A1.8] and A of [A1.9], are introduced in the two equations [A1.3] and [A1.4], finally obtaining the Lorentz transformations: đ?‘Łđ?‘Ľ đ?‘Ąâˆ’ 2 đ?‘Ľ − đ?‘Łđ?‘Ą đ?‘? đ?‘Ľâ€˛ = ; đ?‘Ąâ€˛ = 2 2 √1 − đ?‘Ł2 √1 − đ?‘Ł2 đ?‘? đ?‘? Annex 2 Maxwell equations Maxwell´s first equation The first Maxwell equation [A2.1] indicates that the divergence of the electric field, that is to say the net flow of electric force lines that pass through any closed surface, depends on the density of the electric charge Ď that encloses said surface. If there is no electric charge stored inside the surface, the net flow of the electric power lines that pass through it is zero. 1

⃗∇ . đ??¸âƒ— =

đ?œŒ

[A2.1]

đ?œ€0

E is the electric field, Ď the electric charge density and Îľ0 the dielectric constant in the vacuum or the vacuum polarization capacity. Its value is 8,85∙10-12 C2/N/m2. [A2.1] is deduced from the Gauss´s law, which is the reduced Coulomb’s law for twopoint charges, q1 and Q, separated by a distance r (See the scheme in TABLE A2.1). TABLE A2.1 đ?&#x;? đ???đ??Şđ?&#x;? đ?&#x;’đ?›‘đ?›†đ?&#x;Ž đ??Ť đ?&#x;? F is the force of interaction between the two charges. Applying the concept of field created by a point load, q1, in its environment, we have: đ?&#x;? đ??… đ?&#x;? đ??? đ??„ = đ?&#x;’đ?›‘đ?›† đ??Ş = đ?&#x;’đ?›‘đ?›† đ??Ť đ?&#x;? [A2.2] đ??…=

đ?&#x;Ž

đ?&#x;?

đ?&#x;Ž

The solid angle ΔΊ, subtended by a surface element ΔA, which forms an angle θ with the normal n is: ⃗ ⃗đ?‘&#x;/r2 = ΔAcosθ/r2 ΔΊ = ΔA⃗đ?‘› [A2.3] The total flow of the electric field Δϕ passing through ΔA is, previous substitution of E by [A2.2] and the introduction of the value of ΔΊ of [A2.3]: 1 đ?‘„ 1 ∆∅ = đ??¸âƒ— đ?‘›âƒ—∆đ??´ = đ?‘&#x;đ?‘›âƒ—∆đ??´ = đ?‘„∆đ?›ş 2 4đ?œ‹đ?œ€0 đ?‘&#x; 4đ?œ‹đ?œ€0 Gauss´s law expresses the flow that goes through a spherical surface S in integral form, whose solid angle varies between 0 y 4Ď€ : 4đ?œ‹ ⃗⃗ dA = 1 đ?‘„ âˆŤ đ?‘‘đ?›ş = 1 4đ?œ‹đ?‘„ = đ?‘„/đ?œ€0 Ď• = ∎ ⃗⃗đ??¸ đ?‘› đ?‘†

4đ?œ‹đ?œ€0

0

4đ?œ‹đ?œ€0

The Gauss Ostrogradsky theorem relates the flow through a surface S with the divergence extended to the volume of the closed enclosure V that contains it: đ?œŒ đ?‘‘đ??´ = 1â „đ?œ€ âˆŤ đ?œŒđ?‘‘đ?‘‰ → ⃗∇ . đ??¸âƒ— = [A2.4] ∎ đ??¸âƒ— ⃗⃗⃗⃗⃗ đ?‘

0 �

đ?œ€0

The final result [A2.4] is Maxwell's first equation [A2.1]. Maxwell´s second equation Maxwell´s second equation [A2.5] expresses that all the lines of force of a magnetic field always form a closed loop. Therefore, the divergence of a magnetic field of intensity B is always zero. ⃗∇ . đ??ľ ⃗ =0 [A2.5] [A2.5] confirms the empirical evidence of the absence of magnetic monopoles in nature, or what is the same, the spontaneous emergence of two magnets each time one is split. Maxwell´s third equation Maxwell´s third equation [A2.6] says that any magnetic field that varies with time will cause rotational electric fields ⃗ ⃗∇ Ă— đ??¸âƒ— = − đ?œ•đ??ľ [A2.6] đ?œ•đ?‘Ą [A2.6] is the differential expression of Faraday's law, which establishes that the circulation of an electric field by a closed conductor C is equal and of opposite sign to the

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variation with respect to the time of the magnetic flux traversing a surface S with the contour C as edge (see Figure A2.11): ⃗ ⃗⃗⃗⃗⃗ đ?‘‘đ?‘™ = −đ?‘‘â „đ?‘‘đ?‘Ą âˆŤđ?‘† đ??ľ đ?‘‘đ??´ [A2.7] ∎đ?‘? đ??¸âƒ— ⃗⃗⃗

Figure A2.1 Maxwell´s fourth equation Maxwell´s fourth equation affirms that any electric field that is variable in time produces rotational magnetic fields. ⃗ ⃗ Ă—đ??ľ ⃗ = đ?œ‡0 đ??˝âƒ— + đ?œ‡0 đ?œ€0 đ?œ•đ??¸ ∇ [A2.8] đ?œ•đ?‘Ą

J is the current density and Âľ0 the magnetic permeability in vacuum = 4Ď€âˆ™10-7 N/amp2. [A2.8] is Ampère's empirical law [A2.9] modified. ⃗ ⃗⃗⃗ đ?‘‘đ?‘™ = đ?œ‡0 đ??ź [A2.9] ∎đ??ľ Ampère's Law relates stationary magnetic fields and constant electric currents. It says that the intensity of a magnetic field around a conductor is proportional to the current I that circulates through said conductor (see Fig. A2.22). The difference that equation [A2.8] offers with respect to [A2.9] is due to the term introduced in the second member of [A2.8], which is called the displacement current. This addend was included by Maxwell when he discovered that Ampere's equation, together with Faraday's, broke the principle of conservation of the charge.

Figure A2.2

1 2

Figure taken from https://es.images.search.yahoo.com/search/images. Figure taken from https://upload.wikimedia.org/wikipedia/commons/0/07/Electromagnetism.png.

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However, Maxwell's equations are not enough to summarize all the knowledge of classical electrodynamics. It is necessary to add the equation of the Lorentz force in function of the electric and magnetic fields. ⃗) đ??š = đ?‘ž(đ??¸âƒ— + đ?‘Ł đ?‘‹đ??ľ Annex 2.1 Magnetic field of a linear conductor By Maxwell's equations we arrive at the equation that relates the electric and magnetic fields. E = cB [A2.1.1] By Coulomb’s law E = Q/4πξ0d2 [A2.1.2] By Biot Savart’s law, the magnetic field of a linear conductor is ⃗⃗⃗⃗ đ?‘&#x; đ??źđ?‘‘đ?‘™

Âľ

Âľ

đ?‘‘đ?‘™ đ?‘ đ?‘’đ?‘›đ?œƒ

đ??ľ = 4đ?œ‹0 âˆŤ đ?‘&#x; 2 = 4đ?œ‹0 đ??ź âˆŤ đ?‘&#x; 2 [A2.1.3] |đ?‘&#x;| = 1 , dl is the differential element that creates the magnetic field whose effect B is measured at point P of Figure A2.1.1, located at a distance r. l = d/tgθ; dl = -d/sen2θ dθ [A2.1.4] P

r d

θ dl

l

Figure A2.1.1 On the other hand, r = d/senθ [A2.1.5] Substituting dl and r of [A2.1.4] and [A2.1.5], and the value of the intensity I = Q/t in the integral [A2.1.3], we obtain: Âľ đ?‘„

đ?œƒ

−đ?‘‘

đ?‘‘2

đ??ľ = 4đ?œ‹0 đ?‘Ą âˆŤđ?œƒ đ?‘“ đ?‘ đ?‘’đ?‘›2 đ?œƒ : đ?‘ đ?‘’đ?‘›2 đ?œƒ đ?‘ đ?‘’đ?‘› đ?œƒđ?‘‘đ?œƒ [A2.1.6] đ?‘– Resolving the integral [A2.1.6] between the limits of the initial angle θi and the final angle θf, we get the expression of the magnetic field of a linear conductor. B = Âľ0Q/4Ď€td (cos θi - cos θf) If the linear conductor is an infinite half line, θi = 0 and θf = Ď€/2: B = Âľ0Q/4Ď€td [A2.1.7] Substituting [A2.1.2] and [A2.1.7] in [A2.1.1], we obtain 1/Îľ0 = dÂľ0/t. As c = d/t, we have the ratio of the speed of light as a function of dielectric constant and magnetic permeability in vacuum: 1 đ?‘?= đ?œ€đ?œ‡ [A2.1.8] √ 0 0

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Annex 3 Einstein's general equation of energy The work, W, as a function of the force F and the displacement what it produces, dx, is: đ?‘Ľ1

đ?‘Š = âˆŤ đ??šđ?‘‘đ?‘Ľ đ?‘Ľ0

As F = dp/dt, where p is the momentum, the kinetic energy K will be: đ?‘‘đ?‘? đ??ž = âˆŤ đ?‘‘đ?‘Ą đ?‘‘đ?‘Ľ = âˆŤ đ?‘Łđ?‘‘đ?‘? As đ?‘š0 đ?‘Ł đ?‘‘đ?‘? đ?‘š0 đ?‘? = đ?‘šđ?‘Ł = = 1â „ → 3â „ đ?‘‘đ?‘Ł đ?‘Ł2

{1− 2 } đ?‘?

đ?‘Ł2

2

{1− 2 } đ?‘?

2

[A3.1] [A3.2]

m0 is the rest mass; m, the mass of velocity v and c, the speed of light. Substituting dp in [A3.1] and integrating between v = 0 and v1, speed in which the force stops acting, we get: đ?‘Ł1

đ?‘Ł

đ??ž = âˆŤ0 1 đ?‘Łđ?‘š0

1 3â „ 2 đ?‘Ł2 {1− 2 } đ?‘?

đ?‘‘đ?‘Ł = đ?‘š0

đ?‘?2 1â „ 2 đ?‘Ł2 {1− 2 } đ?‘?

|

= đ?‘š0 0

đ?‘?2 1â „ 2 đ?‘Ł2 {1− 21 } đ?‘?

The summarized form of the equation [A3.3] is: K = (m - m0) c2 E0 = m0c2 E0 is the rest energy. So, K + E0 = E E = mc2

− đ?‘š0 đ?‘? 2

[A3.3]

[A3.4]

Annex 4 Minkowski's four-dimensional space The Minkowski's four-dimensional space has three real dimensions (x, y, z) and an imaginary one (cit). It contains 6 planes: 3 spatial (xy, xz, yz) and 3 temporary spaces (tx, ty, tz). Assuming that the pair of unit vectors in the quadrant (X, cT) of Figure A4.1 are {đ?‘–⃗ , đ?‘—} , and in the quadrant (X ', cT') are {đ?‘˘ ⃗⃗⃗ , đ?‘Ł }, we will have: đ?‘Ľđ?‘– + đ?‘Ąđ?‘— = đ?‘Ľ ′ đ?‘˘ ⃗ + đ?‘Ąâ€˛đ?‘Ł [A4.1]

φ

Figure A4.1 Taking into account the rotation of the axes {X ', cT'} with respect to the axes {X, cT}, the relations between the unit vectors can be established: 5

đ?‘˘ ⃗ = đ?‘?đ?‘œđ?‘ đ?œ‘ đ?‘– + đ?‘ đ?‘’đ?‘› đ?œ‘ đ?‘— đ?‘Ł = −đ?‘ đ?‘’đ?‘› đ?œ‘ đ?‘– + đ?‘?đ?‘œđ?‘ đ?œ‘ đ?‘— Introducing these two expressions in [A4.1], we obtain the coordinates (x, t) in function of (x ', t'). x = x'cos φ - t'sen φ [A4.2] t = x'sen φ + t'cos φ For x' = 0, x = -t'sen φ; t = t'cos φ → tg φ = x/cti = -xi/ct = vi/c As đ?‘ đ?‘’đ?‘› φ =

đ?‘?đ?‘œđ?‘ φ =

đ?‘Ąđ?‘” φ √1 + đ?‘Ąđ?‘”2 φ 1 √1 + đ?‘Ąđ?‘”2 φ

=

=

đ?‘–đ?‘Łâ „ đ?‘? 2 √1 − đ?‘Ł2 đ?‘? 1 2

√1 − đ?‘Ł2 đ?‘? Substituting these two trigonometric functions in [A4.2], we will finally have the Lorentz transformations. đ?‘Łđ?‘Ľ đ?‘Ąâ€˛ + 2 đ?‘Ľ ′ + đ?‘Łđ?‘Ą đ?‘? đ?‘Ľ= ;đ?‘Ą = 2 2 √1 − đ?‘Ł2 √1 − đ?‘Ł2 đ?‘? đ?‘? Dividing x by t we can calculate the speed of a mobile v' with respect to the system (X', cT'), where u is the speed with which this system moves away from the system (X, cT) and v the speed of the mobile with respect to (X, cT). đ?‘Ľ

�′ = � =

đ?‘Ľ ′ +đ?‘Łđ?‘Ąâ€˛ đ?‘Ł đ?‘Ą ′ + 2đ?‘Ľ2 đ?‘?

=

�+� ��

1+ 2 đ?‘?

[A4.3]

For pre-relativistic physics v << c → v' = u + v. On the other hand, in [A4.3] the constancy of the speed of light is fulfilled: if v = c → v' = c. Annex 5 Operators. Metric and quadratic forms of a tensor. Covariant and contravariant of Lorentz 3 Operators We are going to define four operators that act on the components of a vector field VÎą (x) of a four-dimensional spacetime that is fixed at position x: đ?œ• • Derived operator đ?œ•đ?‘Ľ đ?œ‡ : is a linear operator that can be applied to any function (scalar or vector). For example, to the x component of the V vector: đ?œ• đ?›ź đ?œ•đ?‘‰ đ?›ź (đ?‘Ľ) đ?‘‰ (đ?‘Ľ) = đ?œ•đ?‘Ľ đ?œ‡ đ?œ•đ?‘Ľ đ?œ‡ • Multiplier operator, which multiplies one function by another function: f (x)VÎą (x) = f(x). VÎą (x)

3

Much of what is developed in Annexes 5, 6 and 7 is due to the notes taken to Professor Leonard Susskind from Lesson 8 of Stanford University Einstein´s General Theory of Relativity, https://www.youtube.com/watch?v=AC3TMizGpB8.

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For example, the operator đ?‘€đ?›˝đ?›ź đ?‘‰đ?›˝ (đ?‘Ľ) = đ?‘‰ đ?›ź (đ?‘Ľ) is both a matrix and a function. To the left of equality, f (x) is an operator that acts on the column vector VÎą (x); to the right of equality is a product of functions. • Covariant derivative operator âˆ‡Îź: is the derivative with respect to the coordinate of a vector xÎź and acts in the following way: đ?œ• ∇đ?œ‡ = đ?œ•đ?‘Ľ đ?œ‡ + đ?›¤đ?œ‡ [A5.1] đ?›ź Γ is the symbol of Christoffel. It has, for each dimension Îź, a matrix đ?›¤đ?œ‡đ?›˝ with two indices: Îą and β. In the Minkowski's four-dimensional space there are altogether four matrices, one for each dimension. The covariant derivative is a generalization of the partial derivative of vectors that solves the problem of measuring the curvature at a point. Applying the operator on a vector, we obtain: đ?œ•đ?‘‰ đ?›ź (đ?‘Ľ) đ?›ź (đ?‘Ľ)đ?‘‰đ?›˝ (đ?‘Ľ) ∇đ?œ‡ đ?‘‰ đ?›ź (đ?‘Ľ) = + đ?›¤đ?œ‡đ?›˝ đ?œ•đ?‘Ľ đ?œ‡ The first member of this equation is the covariant derivative operator; the first addend of the second member, the derivative operator and the second addend of the second member, the typical numerical matrix that depends on the position x of the vector V. The covariant derivative of a scalar is simply a partial derivative, where the derivation order is indifferent. But the derivation order does affect the result of the covariant derivative applied on a vector. Therefore, it is necessary to resort to the commutator operator. • Commutator operator [AB] = AB - BA đ?œ• Applying the commutator operator [đ?œ•đ?‘Ľ , đ?‘“(đ?‘Ľ)] to vector V, we obtain. đ?œ• đ?œ• đ?œ• đ?œ•đ?‘“ đ?œ•đ?‘‰ đ?œ•đ?‘‰ [ , đ?‘“(đ?‘Ľ)] đ?‘‰(đ?‘Ľ) = đ?‘“(đ?‘Ľ)đ?‘‰(đ?‘Ľ) − đ?‘“(đ?‘Ľ) đ?‘‰(đ?‘Ľ) = đ?‘‰+đ?‘“ −đ?‘“ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ• đ?œ• đ?œ• đ?œ•đ?‘“ [đ?œ•đ?‘Ľ , đ?‘“] = đ?œ•đ?‘Ľ đ?‘“(đ?‘Ľ) − đ?‘“(đ?‘Ľ) đ?œ•đ?‘Ľ = đ?œ•đ?‘Ľ [A5.2] Metric and quadratic forms of a tensor In a plane of dimensions dx1 y dx2 → ds2 = (dx1)2 + (dx2)2, but in space the triangle of decomposition of the vector is spherical. Pythagoras is not fulfilled and the gmn metric4 must be entered in the quadratic form. The quadratic form m dimensional will be: đ?‘‘đ?‘ 2 = ∑ đ?‘‘đ?‘Ľ đ?‘š đ?‘‘đ?‘Ľ đ?‘› = đ?›żđ?‘šđ?‘› ∑ đ?‘‘đ?‘Ľ đ?‘š đ?‘‘đ?‘Ľ đ?‘› đ?‘š

đ?‘šđ?‘›

δmn = Kronecker delta5 = 1, if m = n; δmn = 0, if m ≠n. On the other hand đ?œ•đ?‘Ľ đ?‘š đ?‘&#x; đ?‘š đ?‘‘đ?‘Ľ = đ?‘‘đ?‘Ś đ?œ•đ?‘Ś đ?‘&#x; Substituting dxm in the quadratic form, it results: đ?œ•đ?‘Ľ đ?‘š đ?‘&#x; đ?œ•đ?‘Ľ đ?‘› đ?‘ 2 đ?‘‘đ?‘ = đ?›żđ?‘šđ?‘› ∑ đ?‘&#x; đ?‘‘đ?‘Ś đ?‘‘đ?‘Ś = đ?‘”đ?‘šđ?‘› đ?‘‘đ?‘Ś đ?‘&#x; đ?‘‘đ?‘Ś đ?‘ đ?œ•đ?‘Ś đ?œ•đ?‘Ś đ?‘ đ?‘šđ?‘›

4 5

For spacetime the notation of the metric tensor is g¾ʋ. In the plane, with unit vectors ei, ej, ei.ej = 1, if i = j; ei.ej = 0, if i ≠j.

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Covariant and contravariant of Lorentz Assuming a warped curved surface Ď•, the vertical gradient in the x-y plane has two components: đ?œ•âˆ… đ?œ•âˆ… đ?‘‘∅đ?‘Ľ = đ?‘‘đ?‘Ľ; đ?‘‘∅đ?‘Ś = đ?‘‘đ?‘Ś đ?œ•đ?‘Ľ đ?œ•đ?‘Ś The gradient along the trajectory s will be: đ?œ•âˆ… đ?œ•âˆ… đ?‘‘∅đ?‘ = đ?‘‘∅đ?‘Ľ + đ?‘‘∅đ?‘Œ = đ?‘‘đ?‘Ľ + đ?‘‘đ?‘Ś đ?œ•đ?‘Ľ đ?œ•đ?‘Ś Establishing the change of variables x = x1; y = x2; z = x3..., the general equation of the previous gradient will be: đ?œ•âˆ… đ?œ•âˆ… đ?œ•âˆ… đ?‘‘∅đ?‘ = 1 đ?‘‘đ?‘Ľ1 + 2 đ?‘‘đ?‘Ľ 2 + â‹Ż = ∑ đ?‘› đ?‘‘đ?‘Ľ đ?‘› đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?‘›

If we change the frame of reference of x1-x2 a y1-y2: đ?œ•âˆ… đ?œ•đ?‘Ś 1

=

đ?œ•âˆ… đ?œ•đ?‘Ľ 1

đ?œ•âˆ… đ?œ•đ?‘Ľ 2

đ?œ•đ?‘Ľ 1 đ?œ•đ?‘Ś

đ?œ•đ?‘Ľ 2 đ?œ•đ?‘Ś 1

+ 1

→

đ?œ•âˆ… đ?œ•đ?‘Ś đ?‘›

= ∑đ?‘š

đ?œ•âˆ… đ?œ•đ?‘Ľ đ?‘š đ?œ•đ?‘Ľ đ?‘š đ?œ•đ?‘Ś đ?‘›

[A5.3]

Deriving Ď• with respect to all the components yi, we obtain the transformation equations of a vector V of dimension n located in a reference frame y. đ?œ•đ?‘Ś đ?‘› đ?‘‰đ?‘› (đ?‘Ś) = ∑đ?‘š đ?œ•đ?‘Ľ đ?‘š đ?‘‰đ?‘š (đ?‘Ľ) [A5.4] 6 On the other hand, it is defined the tensor Tmn = AmBn, being A and B vectors. In the plane, m = 2 and n = 2, Tmn has 4 values; en three dimensions, m = 3 and n = 3, Tmn has 9 values, and in the spacetime, m = 4 and n = 4, Tmn has 16 values. AmBn is the product of two vectors of respective dimensions m and n. According to [A5.4], in the frame of reference y: đ?œ•đ?‘Ś đ?‘š đ?œ•đ?‘Ś đ?‘› đ??´đ?‘š (đ?‘Ś)đ??ľđ?‘› (đ?‘Ś) = ∑ đ?‘&#x; đ??´đ?‘&#x; (đ?‘Ľ) ∑ đ?‘ đ??ľđ?‘ (đ?‘Ľ) đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?‘&#x;

đ?œ•đ?‘Ś đ?‘š đ?œ•đ?‘Ś đ?‘›

đ?‘

đ?‘‡đ?‘šđ?‘› (đ?‘Ś) = ∑đ?‘&#x;đ?‘ đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘Ľ đ?‘ đ?‘‡đ?‘&#x;đ?‘ (đ?‘Ľ) [A5.5] [A5.5] is the contravariant transformation of a reference frame tensor x, of dimension rs, in a reference tensor of dimension mn. The inverse transformation is the covariant of the same tensor Tmn (y). đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘Ľ đ?‘ đ?‘‡đ?‘šđ?‘› (đ?‘Ś) = ∑đ?‘&#x;đ?‘ đ?‘š đ?‘› đ?‘‡đ?‘&#x;đ?‘ (đ?‘Ľ) [A5.6] đ?œ•đ?‘Ś

đ?œ•đ?‘Ś

If two tensors are equal, Wnm (x) = Vnm (x) in a reference frame x, they have equals in any frame of reference. What does not mean that if đ?œ•đ?‘‰đ?‘š (đ?‘Ľ) đ?œ•đ?‘‰ (đ?‘Ś) (đ?‘Ś) = đ?‘š đ?‘› đ?‘‡đ?‘šđ?‘› (đ?‘Ľ) = đ?œ•đ?‘Ľ [A5.7] đ?‘› = đ?›ťđ?‘› đ?‘‰đ?‘š → đ?‘‡đ?‘šđ?‘› đ?œ•đ?‘Ś On the contrary, in the reference frame y, the tensor will be [A5.6]: đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘Ľ đ?‘

đ?‘‡đ?‘šđ?‘› (đ?‘Ś) = đ?œ•đ?‘Ś đ?‘š đ?œ•đ?‘Ś đ?‘› đ?‘‡đ?‘&#x;đ?‘ (đ?‘Ľ)

[A5.8]

Substituting in [A5.8] the value of Trs (x) of the first equality of [A5.7], after changing the range of the tensor mn by rs, we will have: đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘Ľ đ?‘ đ?œ•đ?‘‰đ?‘&#x; (đ?‘Ľ) đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘‰đ?‘&#x; (đ?‘Ľ) đ?œ•đ?‘‰đ?‘š (đ?‘Ś) đ?‘‡đ?‘šđ?‘› (đ?‘Ś) = đ?œ•đ?‘Ś đ?‘š đ?œ•đ?‘Ś đ?‘› đ?œ•đ?‘Ľ = ≠đ?œ•đ?‘Ś [A5.9] đ?‘ đ?‘› đ?œ•đ?‘Ś đ?‘š đ?œ•đ?‘Ś đ?‘› which means that Tmn (y) is not a Lorentz scalar, because it depends on the frame of reference. So that it does not depend, we must calculate Tmn (y) from the second equality of [A5.7] and of the expression [A5.10], inverse of [A5.4] đ?œ•đ?‘Ľ đ?‘&#x;

đ?‘‰đ?‘š (đ?‘Ś) = đ?œ•đ?‘Ś đ?‘š đ?‘‰đ?‘&#x; (đ?‘Ľ) 6

A scalar is a tensor of rank 0 and a vector is a tensor of rank 1.

8

[A5.10]

đ?‘‡đ?‘šđ?‘› (đ?‘Ś) = đ?œ•

đ?œ•đ?‘‰đ?‘š (đ?‘Ś) đ?œ•đ?‘Ś đ?‘›

đ?œ•đ?‘Ľ đ?‘&#x;

đ?œ•

đ?œ•đ?‘Ľ đ?‘&#x; đ?œ•đ?‘‰đ?‘&#x; (đ?‘Ľ)

= đ?œ•đ?‘Ś đ?‘› (đ?œ•đ?‘Ś đ?‘š ∙ đ?‘‰đ?‘&#x; (đ?‘Ľ)) = đ?œ•đ?‘Ś đ?‘š

đ?œ•đ?‘Ś đ?‘›

đ?œ•

đ?œ•đ?‘Ľ đ?‘&#x;

+ đ?‘‰đ?‘&#x; (đ?‘Ľ) đ?œ•đ?‘Ś đ?‘› đ?œ•đ?‘Ś đ?‘š

[A5.11]

đ?œ•đ?‘Ľ đ?‘&#x;

đ?‘&#x; đ?›¤đ?‘›đ?‘š = đ?œ•đ?‘Ś đ?‘› đ?œ•đ?‘Ś đ?‘š is Christoffel’s symbol. đ?‘&#x; Introducing in [A5.11] đ?›¤đ?‘›đ?‘š and taking into account that the first addend of the fourth member of [A5.11] is the partial derivative [A5.10] of the vector with respect to yn, we will finally obtain the value of the tensor in the system of reference y. đ?œ•đ?‘‰đ?‘š (đ?‘Ś) đ?œ•đ?‘‰đ?‘š (đ?‘Ś) đ?‘&#x; đ?‘‡đ?‘šđ?‘› (đ?‘Ś) ≠đ?œ•đ?‘Ś = đ?›ťđ?‘› đ?‘‰đ?‘š = đ?œ•đ?‘Ś + đ?›¤đ?‘›đ?‘š đ?‘‰đ?‘&#x; (đ?‘Ľ) [A5.12] đ?‘› đ?‘›

The last addend of the equation [A5.12] is the correction term in a transformation of a tensor when it changes its reference, and đ?›ťn is the covariant derivative, that is, a generalization of the partial derivative in flat space extended to a spacetime curved of n variables. From [A5.12], the covariant derivative of the tensor Tmn is obtained. đ?œ•đ?‘‡ đ?‘&#x; đ?‘&#x; ∇đ?‘? đ?‘‡đ?‘šđ?‘› = đ?œ•đ?‘Śđ?‘šđ?‘› [A5.13] đ?‘? + đ?›¤đ?‘?đ?‘š đ?‘‡đ?‘&#x;đ?‘› + đ?›¤đ?‘?đ?‘› đ?‘‡đ?‘šđ?‘&#x; From the general equation of the quadratic form [A5.14], we can obtain the covariant derivative of the metric gmn. đ?‘‘đ?‘ 2 = đ?‘”đ?‘šđ?‘› đ?‘‘đ?‘Ľ đ?‘š đ?‘‘đ?‘Ľ đ?‘› [A5.14] It was said above that gmn is constant in the plane in all references. So: đ?›ťgmn (x) = 0 [A5.15] Substituting Tmn for gmn in [A5.13], the covariant derivative of the metric gmn is obtained. đ?œ•đ?‘” đ?‘&#x; đ?‘&#x; ∇đ?‘? đ?‘”đ?‘šđ?‘› = đ?œ•đ?‘Śđ?‘šđ?‘› [A5.16] đ?‘? + đ?›¤đ?‘?đ?‘š đ?‘”đ?‘&#x;đ?‘› + đ?›¤đ?‘?đ?‘› đ?‘”đ?‘šđ?‘&#x; = 0 [A5.16] is the relationship between Christoffel symbols and the metric of a tensor. For the above equation to be accomplished, the following relationship must be verified: 1

∂gdc

2

∂xb

đ?‘Ž (x) = g ad { đ?›¤đ?‘?đ?‘?

+

∂gab ∂xc

−

∂gbc ∂xd

}

[A5.17]

Annex 6 Ricci tensor and curvature In the process of translation around the small loop ABCD, the initial position A is not found in the return of the movement, but the point A'. The change of A' for A is due to the presence of curvature in the ABCD plane7 of Figure A6.1. D

C

δxĘ‹ A' A

B dxÂľ

Figure A6.1 We suppose a position vector Va which passes successively to Vb, Vc and Vd. The problem is to identify Va' - Va. [(Vc - Vd) - (Vb - Va)] is the horizontal displacement of the vector; [(Va - Vb) - (Vd - Va')] is the vertical displacement of the vector. 7

The anomaly of not finding the original point of movement is characteristic of spherical or conical surfaces.

9

[(Vc - Vd) - (Vb - Va)] = δxνdxÎźâˆ‡νâˆ‡ÎźV [A6.1] ν Îź [(Va - Vb) - (Vd - Va')] = δx dx âˆ‡Îźâˆ‡νV [A6.2] The covariant derivative, ∇νδxν is the difference experienced by the vector in its travel ABCD with respect to the direction δxν; dxÎźâˆ‡Îź is the difference with respect to the dxÎź direction. Subtracting [A6.1] from [A6.2], we obtain: (Va - Va') = δV = δxνdxÎź [∇ν, âˆ‡Îź]V [A6.3] The expression [∇ν, âˆ‡Îź] of [A6.3] is the commutator of the covariant derivatives. When the initial position A of the circulation ABCD is not found within a loop, the commutator of [A6.3] computes δV. We look for the curvature tensor that tells us how the vector V works in the loop. Applying now the covariant derivative operator [A5.1] in [A6.3], we get [∇ν, âˆ‡Îź] = (đ?œ•đ?œˆ + đ?›¤đ?œˆ )(đ?œ•đ?œ‡ + đ?›¤đ?œ‡ ) − (đ?œ•đ?œ‡ + đ?›¤đ?œ‡ )(đ?œ•đ?œˆ + đ?›¤đ?œˆ ) = đ?œ•đ?œˆđ?œ•đ?œ‡ − đ?œ•đ?œ‡đ?œ•đ?œˆ + [đ?›¤đ?œˆ đ?œ•đ?œ‡ − đ?œ•đ?œ‡đ?›¤đ?œˆ ] + [đ?œ•đ?œˆđ?›¤đ?œ‡ − đ?›¤đ?œ‡ đ?œ•đ?œˆ] + [đ?›¤đ?œˆ đ?›¤đ?œ‡ − đ?›¤đ?œ‡ đ?›¤đ?œˆ ] đ?œ•đ?›¤

The first bracket of the second member is, by [A5.2], equal to [−đ?œ•đ?œ‡đ?›¤đ?œˆ (đ?‘Ľ)] = − đ?œ•đ?‘Ľđ?œˆđ?œ‡ ; the đ?œ•đ?›¤

value of second bracket is [đ?œ•đ?œˆđ?›¤đ?œ‡ (đ?‘Ľ)] = đ?œ•đ?‘Ľđ?œ‡đ?œˆ We can now put [A6.3] in the following way: đ?›ź đ?›ź đ?›ź đ?›ż đ?›ź đ?›ż δxνdxÎź{đ?œ•đ?œˆđ?›¤đ?œ‡đ?›˝ − đ?œ•đ?œ‡đ?›¤đ?œˆđ?›˝ + đ?›¤đ?œˆđ?›ż đ?›¤đ?œ‡đ?›˝ − đ?›¤đ?œ‡đ?›ż đ?›¤đ?œ‡đ?›˝ }đ?‘‰đ?›˝ = đ?›żđ?‘‰ đ?›ź [A6.4] Îź and ν are the main directions of the movement, and β and Îą, the angles of the vector V on the plane ABCD. The displacement suffered by the vector between A and A' is δVÎą, proportional to the size of the rectangle ABCD on which the movement is made in the form of a small loop, that is, proportional to δxνdxÎź. [A6.4] contains the complex structure of the packet of operators and matrices enclosed in curly brackets, a structure known as the Riemann tensor, which acts on the Vβ vector. đ?›ź Therefore, the Riemann tensor đ?‘…đ?œˆđ?œ‡đ?›˝ determines the change of a vector in movement within a curved space in the presence of a small loop. đ?›ź đ?›ź đ?›ź đ?›ź đ?›ż đ?›ź đ?›ż đ?‘…đ?œˆđ?œ‡đ?›˝ = {đ?œ•đ?œˆđ?›¤đ?œ‡đ?›˝ − đ?œ•đ?œ‡đ?›¤đ?œˆđ?›˝ + đ?›¤đ?œˆđ?›ż đ?›¤đ?œ‡đ?›˝ − đ?›¤đ?œ‡đ?›ż đ?›¤đ?œ‡đ?›˝ } [A6.5] From [A6.5] it is concluded: the last two members of the structure enclosed between curly brackets are identical, simply changing the subscript ν by Îź; the indices ν y Îź, as well as Îą and β, are antisymmetric when they are interchanged with each other, and the indexes Îź and β, as well as Ď… and Îą, are symmetrical when they are interchanged with each other. On the other hand, from [A5.17] it follows that the matrix Γ is proportional to the first derivative of the metric tensor, G. ∂Γ therefore contains the second derivative of G, and Γ.Γ are quadratic forms of the first derivative of G. Hence, the Riemann tensor R is defined by containing two kinds of terms: quadratic forms of the first derivative of G and second derivatives of G. A simple and elegant definition formula of the Riemann tensor is: RνΟ = [∇ν âˆ‡Îź] [A6.6] Substituting [A6.6] in [A6.3]: δV = dxÂľdxÂľR¾ʋV [A6.7] It is concluded therefore that the action of the curvature tensor R¾ʋ on the vector V causes the small loop δV. [A6.7] can be put in function of the metric gΟν of a curved space in the following way: đ?›ź đ?›żđ?‘‰ = đ?‘”đ?œ‡đ?œˆ đ?‘…đ?œ‡đ?œˆđ?›˝ [A6.8] As the curvature, by definition, is a scalar, the degree of the Riemann tensor must be lowered to the range two. The procedure consists of taking advantage of the fact that the Riemann tensor has a certain number of symmetries, and that it can be contracted using the metric tensor gΟν to reduce its indices. For example, the Ricci tensor, introduced in 10

1903 by the Italian mathematician Gregorio Ricci, is a symmetric tensor of range two that is obtained from the Riemann curvature tensor. Since space-time has four dimensions, the Ricci tensor does not completely determine the curvature, but when the dimension of the Riemann tensor is reduced below four, the Ricci tensor determines it completely. If the Ricci tensor is contracted a scalar is obtained, known as scalar of curvature R. From the Riemann tensor the Ricci tensor or scalar curvature R is obtained. In [A6.8], the đ?›ź tensor đ?‘…đ?œ‡đ?œˆđ?›˝ is antisymmetric (contravariant) with respect to Îź and Ď…; but the metric g is symmetric (covariant) with respect to Îź and Ď…. When we have a symmetric matrix and it is contracted with an antisymmetric, it is obtained always get zero. The same happens with Îą and β: R is antisymmetric with respect to these two parameters and g is symmetric; any contraction through Îą and β gives zero. But if Îą is contracted with Ď… by adding all the đ?›ź matrices Îą, so that đ?‘…đ?œ‡đ?›źđ?›˝ passes to RΟβ, the Ricci tensor, RΟβ = RβΟ8, is obtained. The Ricci tensor is one of the symmetries of the Riemann tensor, for which the Îź and β indices are symmetric. The scalar curvature, R, is obtained contracting the indices of the Ricci tensor, Îź and β; that is, by means of the operation gβΟRβΟ = R. In Table 6.1, the states of the space are fixed in relation of the values of the scalar R, the Ricci tensor and the Riemann tensor. Table 6.1 Scalar and tensors State of the space Ricci tensor = Riemann tensor = 0 Necessary and sufficient condition of flat space Ricci tensor = Riemann tensor ≠0 Necessary but not sufficient condition of flat space R≠0 Curved space and scalar components Scalar and tensors Dimensions State of the space R = Riemann tensor = 0 Two Flat space Ricci tensor = 0 Three Flat space R = 0 and Ricci tensor = 0 More than Flat space if all three components of the Riemann tensor = 0 R=0

Annex 7 Einstein field equations 9 Newton's equation that expresses the concept of gravity as force is: đ?‘‘2đ?‘Ľ đ??š = đ?‘šđ?‘– 2 = đ?‘šđ?‘” đ?‘”(đ?‘Ľ) đ?‘‘đ?‘Ą

8

The contraction of tensors is an operation that reduces the indexes of a tensor. It is used specially to prove that certain magnitudes are scalars. The contraction of two tensors is done by matching one of the upper indexes of the first tensor with another of the lower indexes of the second tensor, by adding the products of the components over the repeated indexes. This operation automatically decreases the total order of the two tensors. In the case of the đ?›ź contraction of Îą with Ď…, the matched index is Îą, corresponding to the contravariant tensor đ?‘…đ?œ‡đ?œˆđ?›˝ with Ď…, Ου common to the contravariant tensor and the covariant tensor g . 9 Much of what is developed in this Annex is due to the notes taken to Professor Edmund Bertschinger of MIT OpenCourseWare in the lesson Einstein's Field Equations https://www.youtube.com/watch?v=8MWNs7Wfk84.

11

mi is the inertial mass and mg the gravitational mass. The first equation relates force to acceleration; in the second equation, the gravity field đ?‘”(đ?‘Ľ) tells the mass how it accelerates. đ?‘”(đ?‘Ľ) = −đ??şđ?‘šđ?‘– (đ?‘Ľ − ⃗⃗⃗ đ?‘Ľđ?‘– ) G is the gravitational constant and (mi, xi) the point mass and its corresponding position. Gravity as a potential gradient Ď•10 is: đ?‘‘2 đ?‘Ľ ⃗ ; ⃗∅ = (đ?œ•âˆ… đ?‘ĽĚ‚ + đ?œ•âˆ… đ?‘ŚĚ‚ + đ?œ•âˆ… đ?‘§Ě‚ ) = đ?‘” = −∇∅ [A7.1] đ?‘‘đ?‘Ą 2

đ?œ•đ?‘Ľ

đ?œ•đ?‘Ś

đ?œ•đ?‘§

The differential form of the gravitational law is determined by the Poisson equation: ∇2 ∅ = −4đ?œ‹đ??şđ?œŒ(đ?‘Ľ) [A7.2] Ď is the density of the gravitational mass. The integral form of the gravitation law is: đ?œŒ(đ?‘Ľâ€˛)đ?‘‘đ?‘Ľâ€˛ ∅(đ?‘Ľ) = −đ??ş âˆŤ |(đ?‘Ľ) − (đ?‘Ľâ€˛)| Perspective of general relativity: gravity is the curvature of space-time. How spacetime tells the mass how to move? In free fall, the mass moves along a geodesic space-time. The geodesics are extreme own time curves, whose position is a locus of coordinates of xÎź Îź Ń” {0,1,2,3}, where 0 is t, 1, x1, 2, x2 y 3, x3. If a particle moves along a geodesic with a space-time metric, the role of general relativity đ?œ‡ is very simple. Within the geodesic is fulfilled that the tangent vector đ?œ•đ?‘Ľ â „đ?œ•đ?œ? is a constant covariant. If the covariant derivative operator [A5.1] is applied, the equation [A7.3] is obtained. đ?‘‘2 đ?‘Ľ đ?œ‡

đ?œ‡ đ?‘‘đ?‘Ľ đ?œŽ đ?‘‘đ?‘Ľ đ?œ†

+ [đ?›¤đ?œŽđ?œ† đ?‘‘đ?œ? đ?‘‘đ?œ? ] = 0 [A7.3] đ?‘‘đ?œ?2 Îź Ď„ is the proper time, dx /dĎ„, the tangent vector and the term enclosed in brackets is the covariant derivative of the tangent vector. [A7.3] tells us that the particle moves within a gravitational field or in a space-time metric. Therefore, the trajectory that defines equation [A7.3] is along the geodesics. dĎ„2 = g00(dx0)2 +g01dx0dx1+.... [A7.4] 2 gΟν are the coefficients of the metric defined by equation [A7.4]. dĎ„ has 16 components in a 4 x 4 matrix called matrix of metric coefficients or metric. The calculation of the maximum of Ď„ corresponds to the Euler-Lagrange equation: đ?‘‘2 đ?‘Ľ

1

đ?œ•đ?‘”

đ?œ•đ?‘”

đ?œ•đ?‘”

�� � �� �

đ?›źđ?›˝ đ?›źđ?œˆ = − 2 đ?‘”đ?œ‡đ?œˆ ( đ?œ•đ?‘Ľđ?›˝đ?œˆ [A7.5] đ?›ź + đ?œ•đ?‘Ľ đ?›˝ − đ?œ•đ?‘Ľ đ?œˆ ) đ?‘‘đ?œ? đ?‘‘đ?œ? The equation analogous to [A7.5] is [A7.1]. They differ only in that [A7.1] has three components and [A7.5] has four. How does the mass tell spacetime how to curve? In other words, how is the gΟν metric determined? The minimum interval of a geodesic is found when the covariant derivative ∇dxÂľ/dĎ„ = 0. If the covariant derivative operator of the equation [A5.12] is applied, after replacing the vector Vm by dxÂľ/dĎ„, we obtain:

đ?‘‘đ?œ?2

đ?‘‘đ?‘Ľ đ?œ‡

đ?œ• đ?œ•đ?‘ĽÂľ

đ?œ•2 đ?‘Ľ 2

∇ đ?‘‘đ?œ? = đ?œ•đ?œ? đ?œ•đ?œ? + Γ = 0 → đ?œ•đ?œ?2 = đ?‘Žđ?‘?đ?‘’đ?‘™đ?‘’đ?‘&#x;đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› = −Γ [A7.6] Recalling that the acceleration is the gradient of the potential function, according to [A7.1], [A7.6] relates Γand Ď• as follows: đ?œ•âˆ… đ?›¤ = − đ?œ•đ?‘Ľ [A7.7] Equation [A5.17] relates Γ, that is, acceleration, to the metric of the tensor g. On the other hand, Newton's law is fulfilled for slow movements compared to c and for weak 10

For example, if Ď• is the potential energy Ď• = -mgx, the force is the gradient F = −

12

∂∅ ∂x

= −∇F = −mg.

gravitational fields. When that happens, in [A5.17], gad = 1 and the partial derivatives of ∂g00

g are very small, with the only exception of ∂x . Once these substitutions are made, the equation [A5.17] is thus reduced, taking into account [A7.7]: 1 ∂g00

đ?œ•âˆ…

Γ = 2 ∂x = − đ?œ•đ?‘Ľ [A7.8] 00 Integrating [A7.8], it turns out that the value of metric tensor is g = 2Ď• + constant → Ď• = g00/2 [A7.9] We now start from the effect of the Newtonian gravitational field on a radius r on a surface dA: đ??şđ?‘€ âˆŤ đ??šđ?‘‘đ??´ = âˆŤ 2 4đ?œ‹đ?‘&#x; 2 = −4đ?œ‹đ??şđ?‘€ đ?‘&#x; Applying Gauss's divergence theorem: âˆŤ

đ??šđ?‘‘đ??´ = âˆŤ

ĂĄđ?‘&#x;đ?‘’đ?‘Ž

∇đ??šđ?‘‘đ?‘‰ = −4đ?œ‹đ??şđ?‘€

đ?‘Łđ?‘œđ?‘™đ?‘˘đ?‘šđ?‘’đ?‘›

As Ď = M/V, taking into account that the gravity g, and therefore F which is proportional to g, is the gradient of Ď• with opposite sign, according to [A7.1]: ∇đ??š = −4đ?œ‹đ??şđ?œŒ = −∇2 Ď• [A7.10] Introducing [A7.9] in [A7.10], we arrive at the equation đ?›ť2 g00 = 8Ď€GĎ [A7.11] [A7.11] relates the metric g00 to the density of mass Ď , which is really an energy density according to the Einstein equation, E = mc2. Substituting the density of mass Ď 11 for the energy density T00, equation [A7.11] has the following form, which defines the geometric picture of how gravity works: ∇2g00 = 8Ď€GT00 [A7.12] T00 is the energy density element; T0i is the energy flow in the direction i; Ti0 is the density of the momentum in the direction i, and Tij is the rhythm of flow force of the component i per unit area in the direction j, that is, it is the flow rate of the pressure. All these elements are part of the stress-energy or energy-momentum matrix, which is a useful instrument to describe the energy flow and the linear momentum of a continuous distribution of matter. It has 4 x 4 = 16 components according to Table A7.1. The matrix contains in its first row the components of the density of the momentum; in its first column, the momentums, and in the rest the components of the force, including the pressure.

T00 T10 T20 T30

Table A7.1 T01 T02 T11 T12 T21 T22 T31 T32

T03 T13 T23 T33

The problem that arises is that it is not necessary an equation in terms of tensors metric like [A7.12], but of tensors. Substituting the component of the metric g00 from the left of equation [A7.12] by the Einstein tensor GΟν, and the element T00 by the force-energymomentum tensor T¾ʋ, we obtain the Einstein field equations in the space time. GΟν = 8Ď€GTΟν [A7.13]

In equation [A7.11], the source of Newtonian gravity Ď depends on the frame of reference. The energy density T00, which replaces it in [A7.12], also depends on it, but in the relativistic model the energy (E = mc2) is part of the momentum (p = mc), which does not depend on the reference frame. 11

13

The first member of equality [A7.13] refers to the space-time curvature and the second member refers to energy. But the equation presents a problem. Using covariant derivatives as required by relativity, by the conservation of energy ∇TΟν = 0. We need, therefore, on the left side of the equation something whose covariant derivative is zero, so that equation [A7.13] is accomplished. Einstein found that đ?›ťR¾ʋ = 1/2đ?›ťg¾ʋR → (R¾ʋ - 1/2g¾ʋR) = 0 Therefore, the two covariant derivatives, đ?›ťTΟν and (R¾ʋ - 1/2g¾ʋR) are null. Then in equation [A7.13] the tensor GΟν can be replaced by its equivalent, after introducing the factor c4, so that equation [A7.13] keeps the same dimensions. R¾ʋ - 1/2g¾ʋR = 8Ď€G/c4T¾ʋ But for the equation to be universal in all frames of reference, we must impose the condition that the covariant derivative of the metric of the Einstein tensor be zero, that is, ∇gΟν = 0. In this way, Einstein included an additional sum in the above equation. 8đ?œ‹đ??ş đ?‘…đ?œ‡đ?‘Ł − 1â „2 đ?‘”đ?œ‡đ?‘Ł đ?‘… + đ?‘”đ?œ‡đ?‘Ł đ?›Ź = đ?‘? 4 đ?‘‡đ?œ‡đ?‘Ł [A7.14] The first member of [A7.14] refers to the curvature of spacetime; the second, to the mass and energy; Îź and Ę‹ are the dimensions of space time; R¾ʋ, the Ricci tensor, indicating the curvature; g¾ʋ, the metric tensor; R, the scalar of the curvature; Λ, the cosmological constant12 and T¾ʋ, the force-energy-momentum tensor. . Annex 8 Planck length, mass, time and energy Planck length Planck length lp is the distance below which space-time is expected to stop functioning as a classical geometry due to the appearance of quantum gravity. It is the reference scale of string theory. For its calculation, it is necessary to start from the dimensional equations of the speed of light c of the gravitation constant G and of the Planck constant h. The dimensional equation of c is: [c] = LT-1 [A8.1] 8 c = 3∙10 m/s, if c = 1 → 1s <> 3∙108 m. From Newton's law of universal gravitation F = GMm/r2 we can deduce the dimensional equation of G. [G] = MLT-2L2/M2 = L3 T-2M-1 [A8.2] -11 3 2 2 -11 3 2 8 2 2 2 -28 G = 6,7∙10 m /kgs ; G/c = 6,7∙10 m /kgs /(3∙10 ) m /s = 7,4∙10 m/kg Si G = 1 → 1kg <> 7,4∙10-28 m. Finally, from the Planck law E = ħĎ…, [ħ] = ML2T2/T-1 = M L2T-1 [A8.3] -34 2 ħ = 1,1∙10 kg∙m /s; ħ/c = 1,1∙10-34/3∙108 kg∙m2/s/m/s = 3,7∙10-43 kg∙m Si ħ = 1 → 1kg <> 2,7∙1042 m-1. In function of c, G and ħ, the dimensional equation of the Planck length will be [lp] = cÎąGβħγ = L [A8.4]

The cosmological constant Λ was introduced by Einstein to compensate for the effect of the cosmological space. It was proposed as a modification of the original equation of the gravitational field of general relativity for a static universe. Λ had special interest in the wake of the discovery of cosmic acceleration 12

14

Taking into account the dimensional equations [A8.1], [A8.2] and [A8.3], we can present the dimensional equation [A8.4] in matrix form: đ?›ź + 3đ?›˝ + 2đ?›ž 1 1 3 2 (0) = đ?›ź (−1) + đ?›˝ (−2) + đ?›ž (−1) = (−đ?›ź − 2đ?›˝ − đ?›ž ) −đ?›˝ + đ?›ž 0 0 −1 1 β=Îł Îą = -2β - Îł = -3Îł Îą + 3β + 2Îł = 1; -3Îł + 3Îł +2Îł = 1; Îł = β = 1/2; Îą = -3/2. đ??şÄ§

đ?‘™đ?‘? = √ đ?‘? 3 = √

6,7∙10−11 ∙1,1∙10−34 33 ∙1024

√

đ?‘š3 đ?‘˜đ?‘”đ?‘š2 đ?‘˜đ?‘”đ?‘ 2 đ?‘ đ?‘š3 đ?‘ 3

= 1,6 ∙ 10−35 đ?‘š

[A8.5]

Planck time Planck time is the smallest period that can be measured, equivalent to the time it takes for a photon to travel the Planck length at speed c. tp = lp/c = 1,6∙10-35/3∙10-8 tp = 5,4∙10-44 s [A8.6] Planck energy Planck energy is the maximum that can be contained in a sphere of diameter lp. Ep = ħĘ‹ = ħ/tp = 1,1∙10-34 kg∙m2/s/5,4∙10-44 s = 2∙109 kg∙m2/s2 = 2∙109 Nw∙m Ep = 2∙109 J [A8.7] Planck mass Planck mass is the one contained in a sphere of radius lp and that generates the density of the universe of 1093 g/cm3 that the Universe had at the age tp = 5,4∙10-44 s. mp = Ep/c2 = 2∙109 kg∙m2/s2/9∙1016m2/s2 mp = 2,2∙10-8kg [A8.8] Annex 9 SchrĂśdinger equation13 SchrĂśdinger equation describes the wave function Ďˆ (x), whose basic variables are two spatial, (amplitude and wavelength), and one temporal (frequency14). It is assumed that the propagation space of the wave has a certain direction. For example, the wave function [A9.1] propagates in the x direction. Ďˆ = Neikx [A9.1] ⃗ Therefore, the wavelength Îť is in a specific direction, which defines a vector đ?‘˜ through the relationship |k| =2Ď€/Îť. For the development of the SchrĂśdinger equation, we start with three initial postulates: Postulate 1. The wave function is continuous and it admits normalization within a domain.

13

A great deal of what is developed in this Annex 9 is due to the notes taken at MIT open course wave operators and the SchrĂśdinger equations. https://www.youtube.com/watch?v=lMFgfqRZYoc. Instructor: Barton Zweibach. 14 Indistinctly speaking of angular frequency, ɡ = 2πυ, and frequency, Ď… = 1/T, where T is the period.

15

Postulate 2. P (x)dx ≥ Ç€Ďˆ (x)2Ç€dx is the probability of finding Ďˆ (x) in a range dx. Postulate 3. SchrĂśdinger equation allows any linear relationship Ďˆ (x) = ÎąĎˆ1 (x) + βĎˆ2 (x) Operators Operators are instructions on objects telling them how to act. For example, the matrix operator M acts on the components of a vector v indicating the module, direction and ⃗. sense of đ?‘‰ đ?‘Ž11 đ?‘Ž12 đ?‘Ł1 ⃗ đ?‘€đ?‘Ł = (đ?‘Ž ) đ?‘Ł = ( đ?‘Ł2 ) = đ?‘‰ 21 đ?‘Ž22 If the objects are complex functions, the operators act on the functions by means of a code that defines how to apply any function. The following are the fundamental operators of f (x): • Operator đ?&#x;™ identity: f (x) → f (x). đ?œ• đ?œ•đ?‘“ (đ?‘Ľ) • Derivative operator đ?œ•đ?‘Ľ : đ?‘“ (đ?‘Ľ) → đ?œ•đ?‘Ľ . • Multiplier operator Ě‚đ?œ’: f (x) → đ?œ’f (đ?‘Ľ). • Square operator Ě‚ đ?‘†đ?‘ž : f (x) → (đ?‘“ (đ?‘Ľ))2. Ě‚ • Constant operator đ?‘ƒ 42 : f (x) → 42. Ě‚ (x) • Operator Ѳâ„Ž (đ?‘Ľ) : f → h (x)f (x). Ě‚ Of the five finite operators, all are linear except Ě‚ đ?‘†đ?‘ž đ?‘Ś đ?‘ƒ 42 . For example: Ă”(af (x) + bg (x)) = a Ă”f (x) + b Ă”g (x) Postulate 3 states that the SchrĂśdinger equation allows any linear relationship. Therefore, linear operators become necessary. Eigenvectors, which are special vectors whose action on any arbitrary vector keeps it in the same direction, are also very necessary. In general, the action of an operator, such as a matrix, on any vector makes it jump to another position, marking a different direction to the primitive. But an eigenvector matrix of dimension m x n acts through the n rows maintaining the same direction of the vector. After the action of an eigenvector, the magnitude of the vector can grow or decrease, but it keeps the same direction. The proportionality constant of the action on the vector is called eigenvalue. Also, operators can be eigenfunctions. Ă‚ fa (x) = afa (x) [A9.2] fa(x) is the eigenfunction and the eigenvalue a is a number that can take infinite values. The operator Ă‚ does not comply with the additive property. If Ă‚ f1 = a1f1 and Ă‚ f2 = a2f2 → Ă‚ (f1 + f2) = a1f1 + a2f2 Ă‚ is not necessarily a eigenfunction. On the other hand, if fa is an eigenfunction 3fa is the same eigenfunction. Applying the eigenfunction operator momentum đ?‘?Ě‚ to the wave function [A9.1], we obtain: đ?‘?Ě‚ eikx = ħkeikx ; fa (x) = eikx; a = ħk [A9.3] ikx According to [A9.3], e is an eigenfunction and ħk is an eigenvalue. SchrĂśdinger equation SchrĂśdinger equation can be constructed by creating relationships between operators. For example, let's relate the operator momentum as follows: 16

đ?‘?Ě‚ =

ħ đ?œ•

[A9.4]

đ?‘– đ?œ•đ?‘Ľ

The expected value of p will be: ∞ ħ đ?œ• <p> = âˆŤâˆ’âˆž Ďˆâˆ— (x)dx đ?‘– đ?œ•đ?‘Ľ Ďˆ (x) Ďˆ*(x) is the conjugate function of Ďˆ (x). Momentum operator From [A9.4], the momentum operator đ?‘?Ě‚ Ďˆ relates to it in the following way: ħ đ?œ• đ?‘?Ě‚ Ďˆ = đ?‘– đ?œ•đ?‘Ľ Ďˆ (x) = ħkĎˆ; đ?‘?Ě‚ = ħk

[A9.5]

Energy operator We relate the energy operator through the following relationship: đ?‘?Ě‚2

ħ2 đ?œ•2

ĂŠ = 2đ?‘š + đ?‘‰ (đ?‘ĽĚ‚) = − 2đ?‘š đ?œ•đ?‘Ľ 2 + đ?‘‰ (đ?‘Ľ)

[A9.6]

Postulate 4. For each observer there is an associated operator Ă‚ ≥ {for the momentum ↔ đ?‘?Ě‚ ; for the position ↔ đ?‘ĽĚ‚; for energy ↔ ĂŠ} It is absolutely essential to understand that when an operator acts on a function the result of the action has nothing to do with the measure associated with the observable. Expected value of A: <A> = âˆŤ Ă‚ Ďˆ (x)Ďˆâˆ— (x)dx The uncertainty principle acts through the operator. We define as uncertainty the operator: (∆đ??´)đ?œ“ = √< Ă‚2 > −< Ă‚ >2 [A9.7] Operators, like matrices, are not necessarily commutative. For example, ħ đ?œ• (đ?‘?Ě‚ đ?‘ĽĚ‚)đ?‘“ (đ?‘Ľ) ≠(đ?‘ĽĚ‚đ?‘?Ě‚ )đ?‘“ (đ?‘Ľ); (đ?‘?Ě‚ đ?‘ĽĚ‚)đ?‘“ (đ?‘Ľ) ≥ đ?‘?Ě‚ (đ?‘ĽĚ‚đ?‘“ (đ?‘Ľ)) = đ?‘?Ě‚ (đ?‘Ľđ?‘“ (đ?‘Ľ)) = (đ?‘Ľđ?‘“ (đ?‘Ľ)) = đ?‘– đ?œ•đ?‘Ľ ħ đ?‘–

ħ đ?‘Ľđ?œ•đ?‘“ (đ?‘Ľ)

đ?‘“ (đ?‘Ľ) + đ?‘–

đ?œ•đ?‘Ľ

[A9.8] ħ đ?œ•đ?‘“ (đ?‘Ľ)

(đ?‘ĽĚ‚đ?‘?Ě‚ )đ?‘“ (đ?‘Ľ) ≥ đ?‘ĽĚ‚(đ?‘?Ě‚ đ?‘“ (đ?‘Ľ)) = đ?‘ĽĚ‚ đ?‘–

đ?œ•đ?‘Ľ

ħ

= đ?‘–đ?‘Ľ

đ?œ•đ?‘“ (đ?‘Ľ) đ?œ•đ?‘Ľ

[A9.9]

So [A9.8] ≠[A9.9] → (đ?‘?Ě‚ đ?‘ĽĚ‚)đ?‘“ (đ?‘Ľ) ≠(đ?‘ĽĚ‚đ?‘?Ě‚ )đ?‘“ (đ?‘Ľ) Definition of commutator: [Ă‚, ĂŠ] ≥ AE - EA. Subtracting [A9.9] from [A9.8], we obtain: ħ (đ?‘ĽĚ‚đ?‘?Ě‚ − đ?‘?Ě‚ đ?‘ĽĚ‚)đ?‘“ (đ?‘Ľ) = − đ?‘“ (đ?‘Ľ) = đ?‘–ħđ?‘“ (đ?‘Ľ) = [đ?‘ĽĚ‚, đ?‘?Ě‚ ] = Commutator of x with p đ?‘– [xĚ‚, pĚ‚] = iħ[đ?&#x;™] [A9.10] From the equation [A9.10] it deduced: • Since the commutation of x with p gives an imaginary value, x and p are encapsulated, that is, the position and momentum of a particle cannot be measured alternately. • The combination of operators xĚ‚, pĚ‚ is the identity operator. The first conclusion is equivalent to the Heisenberg uncertainty principle. From the second conclusion it follows that the position x and the momentum p provide all the complete information of the evolution of a quantum system over time. If one tries to develop the equation [A9.10] by means of matrices, there is no dimension that validates the operation: neither matrices of 2 x 2, nor of 3 x 3, nor of 4 x 4, etc. Therefore, matrices cannot be computed as the operator [xĚ‚, pĚ‚] except matrices of infinite 17

dimension. This impossibility of establishing matrices that materialize the calculation of the equation [A9.10] is the mathematical manifestation of quantum mechanics. Postulate 5: Once an average is established, and observer A is associated with the operator Ă‚, two things to take into account occur: If the measured quantity is the momentum, the energy or the position, the • measured value is a number (the eigenvalue 15 of Ă‚). The eigenvalues must be real numbers, since the special operators of real • eigenvalues have a real value. After a measurement, the system collapses in the auto-function Ďˆa. Substituting in [A9.2] fa for Ďˆa, we can apply the eigenvalue to Ďˆa in the following way: Ă‚Ďˆa = aĎˆa [A9.11] Suppose now that we look for the position of Ďˆ (x) and that when measuring a particle is found at position x0 Since the measure is one of the eigenvalues a Ç€ Ă‚Ďˆa = aĎˆa, our eigenstate is a particle a in a place. What is the best function associated with an eigenstate position? That is, what is the best eigenfunction Ďˆa | a is in the measurement interval? It is the Dirac delta function 16 δ (x - x0), which is defined at some point and nowhere else, which has a peak at x0 and which is zero at any x ≠x0. We apply the eigenfunction operator according to the equation [A9.11], but now xĚ‚ is the operator, δ (x - x0) is the eigenfunction of x y x0 the eigenvalue of x. So: Ă‚Ďˆa = đ?‘ĽĚ‚δ (x - x0) and aĎˆa = x0δ (x - x0) → Ďˆ (x) ≃ đ?‘ĽĚ‚δ (x - x0) = x0δ (x - x0) Really the functions δ are not used for discontinuous intervals, but for continuous applications such as in the integrals. So Ďˆ (x) ≃ âˆŤ đ?‘ĽÎ´ (x − x0 ) = đ?‘Ľ0 đ?›ż (đ?‘Ľ − đ?‘Ľ0 ) since Ďˆ (x) = 0 except for x = x0. Postulate 6. It has been said above that in the eigenfunction Ďˆa (x), a takes several values. It is now a matter of expanding Ďˆa in terms of eigenfunctions in the same way that Fourier expanded the eigenfunction of the eipx momentum. The expansion requires a previous normalization, by means of orthogonal eigenfunctions of the type: đ?›żđ?‘Žđ?‘? = âˆŤ Ďˆ(x)Ďˆâˆ— (x)dx Likewise, in the expansion should be taken into account a couple of considerations: • Since any state can be formulated as a superposition of other states Ďˆi, Ďˆ (x) can be expanded as đ?œ“(đ?‘Ľ) = ∑ đ??śđ?‘Ž đ?œ“đ?‘Ž (đ?‘Ľ) [A9.12] Ca is a coefficient. • The probability of obtaining the particular value a0 in one measurement is Ç€Ca0Ç€2 = PĎˆ(a0) [A9.13] Therefore, by expanding the wave function Ďˆ (x) in terms of eigenfunctions Ďˆa (x), the square of the coefficient of expansion Ca0 is the probability of obtaining the measurement a0. Example: For the operator xĚ‚, {δ (x - x0) ∀đ?‘Ľ0 } are eigenfunctions If we apply the superposition

15

We recall that the eigenvalue is the constant of proportionality of the action on the operator Ă‚. Dirac Delta is a density function of an idealized point mass, whose value is equal to zero anywhere except for x0 and whose integral in the interval (x - x0) = 1. 16

18

đ?œ“(đ?‘Ľ) = âˆŤ Ďˆ(x0 )δ(x − x0 )dx0 and we compare this equation with [A9.12], it results: Ďˆ(x0) = Ca y δ (x - x0) = Ďˆa. We return to the SchrĂśdinger equation Postulate 7. The SchrĂśdinger equation was based on experimentation. From the Planck equation E = ħɡ, the quantum theory was developed, confirmed by the works of Planck and Einstein, by the Millikan experiment and also by the Compton test of the inelastic scattering of a photon by an electron. The photon is defined by the pair of parameters of energy and momentum. ⃗} {E = ħɡ; đ?‘? = ħđ?‘˜ [A9.14] ⃗ is the wave vector of module 2Ď€/Îť. đ?‘˜ [A9.14] can be condensed as follows ⃗) (đ??¸, đ?‘?) = ħ(ɡ, đ?‘˜ Âľ Four dimensions appear in this equation: in the p component of (đ??¸, đ?‘?), Îź = 0, 1, 2, 3; in the xÂľ component of (đ?‘Ą, đ?‘Ľ), Îź = x1, x2, x3. Equation [A9.14] worked experimentally only for photons, but De Broglie had a bold idea: he extrapolated the quantum equations of the momentum and the energy for any particle, that is to say he posed that the particles in general behaved like waves. He wrote the equation [A9.15] for a wave moving in the increasing sense of t. Ďˆ (x, t) = ei(kx - ɡt) [A9.15] Applying derivatives to the equation [A9.15]: đ?œ•đ?œ“ đ?œ•đ?œ“ = −đ?‘–ɡđ?œ“; đ?‘–ħ đ?œ•đ?‘Ą = ħɡ đ?œ•đ?‘Ą đ?œ•

As E = ħɡ → ĂŠ = đ?‘–ħ đ?œ•đ?‘Ą [A9.16] From the first equality of the equation [A9.5] ħ đ?œ• đ?‘?Ě‚ Ďˆ = đ?‘– đ?œ•đ?‘Ľ [A9.17] We therefore have that applying the operators [A9.16] and [A9.17] to the wave Ďˆ of equation [9.15], it is fulfilled: ħ đ?œ• đ?œ•đ?œ“(đ?‘Ľ,đ?‘Ą) đ?œ“(đ?‘Ľ, đ?‘Ą) = đ?‘?đ?œ“(đ?‘Ľ, đ?‘Ą); đ?‘–ħ = đ??¸đ?œ“(đ?‘Ľ, đ?‘Ą) [A9.18] đ?‘– đ?œ•đ?‘Ľ đ?œ•đ?‘Ą In the first equality of [A9.18] time plays no role, but yes in the second equality. The second equality of [A9.18] tells us how energy evolves as a function of time. E is a scalar that is known how it evolves over time, but does not know what it is. To solve this problem, we try to change E → ĂŠ in the second equality of [A9.18]. Through these two premises is arrived at the SchrĂśdinger equation for every energy-carrying particle. Changing E (a scalar) by ĂŠ (an operator) in the second equality of [A9.18], we get to the general equation: đ?œ•đ?›š(đ?‘Ľ,đ?‘Ą) đ?‘–ħ đ?œ•đ?‘Ą = ĂŠđ?›š(đ?‘Ľ, đ?‘Ą) [A9.19] At the beginning of the Operators section it is said that the operator indicates how the object on which it intervenes acts; in our case, energy. To know the complicated potential in which the particle moves, we apply [A9.6] on Ďˆ (x, t) previous substitution of the operator ĂŠ [A9.19]. đ?œ•đ?›š(đ?‘Ľ,đ?‘Ą)

ħ2 đ?œ•2

đ?‘–ħ đ?œ•đ?‘Ą = − 2đ?‘š đ?œ•đ?‘Ľ 2 đ?œ“(đ?‘Ľ, đ?‘Ą) + đ?‘‰(đ?‘Ľ)đ?œ“(đ?‘Ľ, đ?‘Ą) [A9.20] [A9.20] is a differential equation of first order with respect to t and of second order with respect to x with the following characteristics: linear equation; deterministic (if you know 19

Ďˆ (x, 0) you can calculate Ďˆ (x, t)); of complex variables, and that if it satisfies a wave function as a solution, it also satisfies the sum of two wave functions. The first addend of the second member of [A9.20] is the kinetic energy of the particle (function of time); the second member is the potential energy, where V (x) is the potential field that confines the particle (function of space); the first member is a function of time Putting otherwise the equation [A9.20] it results ħ2

đ?œ•đ?›š

đ?‘–ħ đ?œ•đ?‘Ąđ?‘Ą = − 2đ?‘š đ?›ť 2 đ?œ“đ?‘Ą + đ?‘ˆđ?œ“đ?‘Ą [A9.21] 2 U = u (đ?‘&#x;) is a gradient with respect to the radius r and ∇ = Δ is the Laplacian operator. Comparing the expression [A9.21] with the heat conduction equation: đ?œ•đ?‘‡(đ?‘Ľ,đ?‘Ą)

đ?œ•2

Îą đ?œ•đ?‘Ą = đ?œ•đ?‘Ľ 2 đ?‘‡(đ?‘Ľ, đ?‘Ą) + đ??´đ?‘‡(đ?‘Ľ, đ?‘Ą) The structural identity of both equations is observed if the wave function Ďˆ is replaced by the temperature T. The only element that distinguishes them is the imaginary unit i, which transforms the scalar T into a wave function Ďˆ. This transformation makes Ďˆ an unobservable function, because the first member of [A9.21] is imaginary, and if we were able to solve the equation it would give us a complex number, and we do not know what that means. To avoid the handling of complex variables, it is necessary to appeal to the product of the wave function by its conjugate ĎˆtĎˆt*, which is a real value. In doing so, we renounce to find the ideal solution of the wave function Ďˆ (x, t), which places a particle at a defined location and time. ĎˆtĎˆt* is interpreted as the density function of the probability of finding the particle in its position x and time t. Also, we normalize the function constituted by the product of the wave function by its conjugate, which means that the scale of the potential position of the wave is reduced to 1. Therefore, the probability of locating the wave function is integrated on the full domain space considered ∀. âˆŤ đ?œ“đ?‘Ą đ?œ“đ?‘Ąâˆ— đ?‘‘∀ = 1 With probability, uncertainty arises. The operation with discrete variables of the quantum model is the cause of everything being uncertain: space, momentum, energy and time. In all quantum variables we must therefore calculate the most probable value or expected value, and the associated uncertainty. For a variable q and an operator Ί, which acts on the wave function Ďˆ, the general expression of the expected value is: <q> =âˆŤ đ?œ“đ?‘Ąâˆ— đ?›şđ?œ“đ?‘Ą đ?‘‘∀ [A9.22] The operator Ί of the position is đ?‘&#x;; the operator Ί of momentum is, according to [A9.4], ħ đ?œ• đ?‘?Ě‚ = đ?‘– đ?œ•đ?‘Ľ = −đ?‘–ħ∇ [A9.23] đ?œ•

đ?œ•

đ?œ•

⃗ is another operator. ∇= đ?œ•đ?‘Ľ đ?‘– + đ?œ•đ?‘Ś đ?‘— + đ?œ•đ?‘§ đ?‘˜ The operator Ί of energy is, after substitution in [A9.6] of the potential V (x) in the direction of the x axis by the potential U = u (đ?‘&#x;) in the radial direction r. đ?‘?Ě‚đ?‘?Ě‚ ĂŠ = 2đ?‘š + đ?‘ˆ [A9.24] Introducing the operator p of [A9.23] in [A9.24], it remains: ħ2

ĂŠ = − 2đ?‘š ∇2 + đ?‘ˆ According to the general expression [A9.23], the expected value of E will be: ħ2

∞

<E> = âˆŤâˆ’âˆž Ďˆâˆ— (x)dx(− 2đ?‘š ∇2 + đ?‘ˆ) Ďˆ (x) ∇2 is the Laplacian and U, a gradient. Therefore, as Δ = f (x, y, z) y U = u (đ?‘&#x;)

20

[A9.25]

the solution of [A9.25] is not a function of t. So, we must solve the SchrĂśdinger equation doing separation of variables: đ?œ“đ?‘Ą (đ?‘Ą, đ?‘&#x;) = đ?œ“(đ?‘&#x;âƒ—âƒ—âƒ—â€˛ ) ∙ đ?‘Œ(đ?‘Ą) = đ??¸ [A9.26] E is constant because it is the product of separate variables, where there can be no products x.t. Deriving [A9.26] with respect to t, đ?œ•đ?›šđ?‘Ą đ?‘‘đ?‘Œ đ?›šđ?‘Ą đ?‘‘đ?‘Œ =đ?œ“ = đ?œ•đ?‘Ą đ?‘‘đ?‘Ą đ?‘Œ đ?‘‘đ?‘Ą and substituting this value in [A9.21], we have: ħ2

1

1 đ?‘‘đ?‘Œ

[− 2đ?‘š ∇2 đ?œ“ + đ?‘ˆđ?œ“] = đ?‘–ħ đ?‘Œ đ?‘‘đ?‘Ą đ?œ“

[A9.27]

The first member of [A9.27] depends only on space; the second member depends only on time. It is therefore an equation of separate variables. As in the case of [A9.26], the two members of [A9.27] are equal to the constant E. Y (t) is solved as a linear differential equation Y(t) = Ce-iEt/ħ = Ce-iɡt, since E = hĎ… = hɡ/2Ď€ → ɡ = E/ħ The first member of equation [A9.27] stays as follows: ħ2

− 2đ?‘š ∇2 đ?œ“ + (đ?‘ˆ − đ??¸)đ?œ“ = 0 [A9.28] which is the SchrĂśdinger equation dependent on the position of the particle and independent of t. For different potentials of U there are different states of energy If the operator ĂŠ of [A9.26] is substituted in the equation [A9.28], it is discovered that ĂŠ = E. Therefore, <E> = E and E represents the energy of the system. To clear E from [A9.28], we act in the same way as in the case of heat conduction: solving a differential equation of the second order, for which solution it is necessary to previously know the environmental conditions of the equation. The unknown of the equation is the eigenvalue E, which is equivalent to the quantum potential restriction, but not to the total energy of the system. BIBLIOGRAPHY https://es.images.search.yahoo.com/search/images. https://upload.wikimedia.org/wikipedia/commons/0/07/Electromagnetism.png. https://www.youtube.com/watch?v=lMFgfqRZYoc. Instructor: Barton Zweibach. https://www.youtube.com/watch?v=8MWNs7Wfk84 https://www.youtube.com/watch?v=AC3TMizGpB8

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