Tensors and Special Relativity

Transcription

1 Tensors and Special Relativity Lecture 6 1 Introduction and review of tensor algebra While you have probably used tensors of rank 1, i.e vectors, in special relativity, relativity is most efficiently expressed in terms of tensor algebra. General relativity, however, requires tensor algebra in a general curvilinear coordinate system. Before discussing special relativity, it will be useful to introduce some of the mathematics of differential forms in a general curvilinear set of coordinates, although we eventually develop the needed formulation in a Cartesian coordinates. 1.1 Generalized coordinates All coordinate systems are defined relative to a Cartesian set of coordinates. For 3-D (x 1, x 2, x 2 ), although extension to more spatial dimensions is trivial. Thus there are 3 functions that define the space; ζ i (x 1, x 2, x 3 ) i = 1, 2, 3 We also presume that there exists a unique inverse; x i (ζ 1, ζ 2, ζ 3 ) i = 1, 2, 3 Now at the intersection of the planes; ζ i = constant i i = 1, 2, 3 We define a set of unit vectors, â i, perpendicular to each surface. If these vectors are mutually orthorgonal, an orthorgonal coordinate system is defined. The direction cosines with respect to the Cartesian unit vectors are; â 1 ˆx i = α i â 2 ˆx i = β i â 3 ˆx i = γ i For an orthorgonal system the direction cosines are related, as may be shown by calculating â i â j for i, j = 1, 2, 3. 1

2 3 αi 2 = 1 1=1 3 1=1 3 1=1 and; β 2 i = 1 γ 2 i = 1 α i βiγ 1 = 0 α i γiβ 1 = 0 β i γiα 1 = 0 Now consider the differential element of length, ds. In the Cartesian system, the square of this element is ds ds = 3 dx 2 i Suppose we now introduce a general curvilinear set of coordinates as defined above. dx i = 3 j=1 ζ j dζ j The square of the length element is then ds 2 = 3 3 j,k=1 This is rewritten as ; g jk = 3 ζ j ζ k ζ j ζ k dζ j dζ k where g jk are the metric elements that define the space. Therefore; ds 2 = jk g jk dζ j dζ k h 2 j = i In the case of an orthorgonal system g jk = 0 if j k. Therefore we define a scale factor, ( ζ j ) 2 Note that h j dζ j is the length element for the j th coordinate. Therefore 2

3 a^ 2 α a^ 1 d ζ 1 dh 2 dζ 1 d ζ 2 h 2 dζ 2 a^ 1 h 1 dζ 1 a^2 a^ 1 Figure 1: A cross section of an area element in a generalized coordinate system ds 2 = i (h i dζ i ) 2 and we are able to obtain the volume and area elements as dτ = (h 1 dζ 1 )(h 2 dζ 2 )(h 3 dζ 3 ) dσ k = (h i dζ i )(h j dζ j ) with cyclic permentations of i, j, k. In general h i varies at each point in the coordinate space. The direction cosines along the new coordinate axes (ONLY for an orthorgonal system) are; γ ni = (1/h n ) ζ n = h n ζ n ζ i Not only do the scale factors change with position, but also the unit vectors change directions, Fig. 1. For example, â 1 ζ 2 = ζ 2 i ˆx i h i ζ 1 Which can be reduced to; â 1 ζ 2 = â2 h 1 h 2 ζ 1 It is then interesting to apply these equations to a familiar coordinate system. Use as an example spherical coordinates; x = rcos(φ)sin(θ) 3

4 y = rsin(φ)sin(θ) x = rcos(θ) Take the partial derivatives to show an orthorgonal system is produced and the square of the metric length is; ds 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 (θ)dφ 2 as expected. Unit vectors, and volume/area elements are also evaluated as expected. Also the vector operations gradient, div, and curl can be obtained from the physical definition of these operators. 1.2 Tensors Tensors are introduced by considering the transformation properties of a function under a coordinate rotation and reflection. Thus a scalar function does not change value under rotation or reflection. As an example the function f = 3 (x i x 0,i ) 2 remains constant and is considered the magnitude of a vector. On the other hand if we consider; f = 3 f ˆx i then f transforms as a vector which preserves magnitude but changes direction. It changes sign upon reflection x i x i as immediately seen. A scalar does not change sign upon reflection. If the function is constant under rotation but changes sign under reflection it is a pseudo-scalar. If a vector does not change sign under reflection it is a pseudo-vector. A pseudo-vector can be obtained through the cross product of 2 true vectors. We generalize this description of functions by defining a scalar as a tensor of rank 0 and a vector as a tensor of rank 1. Now we expand the notation to higher rank and discuss the algebra. To help with notation we will now employ the summation convention unless it leads to ambiguities. The summation convention suppresses the symbol and it is understood that if an index is repeated it is summed. Thus i x k x i x i x j = x k x i x i x j Suppose an n-dinemsional space, with N independent variables x i i = 1,, n. The set of x i define a point in this space. Now define a set of n linearly independent functions ζ i (x 1,, x n ) i = 1,, n. For the functions to be linearly independent the Jacobian of the functions does not vanish; 4

5 ζ 1 ζ n x 1 x 1 0 ζ 1 ζ n x n x n The functions define a new coordinate system. Make the substitution x i = ζ i, and then look at k k x j k k = δ j = x k x j x i i x j We also find that dx i = x i x j dx j Now we see there that the differential quantities dx i and dx j are related by a linear transformation, x i x j. A tensor will be described by linear transformation relations between coordinate systems. 1.3 Covarient and contravarient notation We have not distinguished between various transformation properties. Up to this point we have described transformational properites associated with contravarient tensors( eg real vectors). We will write these tensors with a superscript (ie dx i will be written dx i ) to denote it is contravarient. If the element is written with a lower subscript is will be identified as covarient. thus any component that transforms as; A i = x i x j Aj is a contravarient vector. Now consider the differential f f ˆx i This transfrorms as; f x i = f x j xj x i Obviously the transformation is xj x i rather than x i j. This transforms as a covarient x vector. 5

6 A i = x i x j A j The superscript indicates a contravarient component and the subscript indicates a convarient quantity. Thus suppose we write; x k x i A i = xk x i x i x j A j = A k Now observe that if we apply the above transformation as; A i B i = A J x,i x j B k xk x j = A j B j which is an invarient under the transformation. This is defined as a contraction of (reduction in) the tensor order and in this case it represents a scalar (dot) product of 2 vectors. Now 2 contravarient vectors can be directly multipled to produce a tensor of higher rank, in this case a rank 2 tensor. A i B j = x i x k x j x l A k B l Thus write A i B j = a jk. Multiplication of covarient and contravarinet vectors gives tensors of various transformation properties. Therefore tensors of any rank can be produced or contracted by appropriate multiplications and sums. The order of the indicies is important. If the tensor is of one type and remains invarient under the exchange of indicies, then the tensor is symmetric. If the tensor changes sign under the interchange of indicies then the tensor is anti- (or skew) symmetric. Any tensor may be expanded in a sum of symmetric and anti-sumetric tensors. For a second rank tensor this representation takes the form; a ij = (1/2)[a ij + a ji ] + (1/2)[a ij a ji ] Addition of tensors that are not of the same order does not produce another tensor. 2 Conjugate tensors Let g ij be a covarient, symmetric tensor of 2 nd order. Thus g ij = g ji. Denote the determinant of the matrix formed by this tensor as g g 11 g 1n g = g n1 g nn We let g ij be the cofactor of g ij /g. The cofactor of the element a ij is equal to ( 1) i+j M ij where M is the minor of a ij as obtained from the determinant. By this technique we can 6

7 raise of lower an index (change a tensor from covarient to contravarient, etc). Thus note; A i = g ij A j g ki A i = g ki g ij A j g ki A i = δ k j Aj 3 The metric tensor As previously, the square of the length element is; ds 2 = dx i dx i = g jk dx j dx k The g ij form a tensor of 2 nd rank called the metric tensor of the space. We understand that the determinant, g = g ij 0 but it is possible in general to have ds 2 < 0. As this would not be consistent with length, the measure of the space is taken as the absolute value of ds 2. Note that ds 2 is a tensor of rank 0, ie a scalar quantity. 4 Levi-Civita tensor It wil be useful to define the following tensor of rank 3 or higher. ǫ ijk = 1 For i, j, k = 1, 2, 3 if i j k and are an even permutation of 1, 2, 3. The tensor equals -1 if the indicies are an odd permutation of 1, 2, 3 and the tensor is 0 if any of the indicies have the same value. The tensor is a pseudo-tensor, a tensor of inverting symmetry upon interchange of indicies. A conjugate tensor with the same properties can also be defined. The contracton of a pseudo-scalar tensor tensor with another tensor produces another pseudotensor perhaps a pseudo-scalar. This will lead to dual tensors which will be defined later. 5 The postulates of special relativity Observational astronomy and perhaps more precise measurements of physical phenonmena in the local environment of the earth show no effects on the spatial position where the measurement is made. Thus the conclusion that in absence of massive objects, space/time 7

8 x x Vo z z Figure 2: Reference frames for the Lorentz transformation positions have no effect. Thus one first assumes; 1) There exist a set of reference frames moving at constant velocity with respect to each other in which all phenonmena occur in an identical manner. These are inertial reference frames in which Newtonian mechanics holds, eg the law of inertia F = m a. That there is a set on unique frames of reference, is unsatisfactory and corrected in general relativity. 2) The second observation is that the velocity of EM radiation in vacuum is constant, independent of the source or reference frame. We now consider the first postulate. Suppose 2 inertial reference frames as shown in Fig. 2. In transforming between these frames, the relations are; x = x y = y z = z v 0 t t = t The above are the Galilean transformations. The velocity transformations are ; V x = V x 8

9 V y = V y V z = V x V 0 The acceleration transformations are ; a = a which indicates that the law of inertia is the same in all inertial frames of reference. But Maxwell s equations are not invarient under these transformations. Try putting the transformations into the wave equation; V x = V x 2 x 2 (1/c 2 ) 2 t 2 = 0 Maxwell s equations require the 2 nd postulate, above. 6 Simultaneous events Before applying the constancy of the velovity of light, we consider its meaning. In fact, we postulate that there is a limiting velocity, c, and nothing travels faster than this velocity. Therefore, information cannot propagate faster than c. It happens that EM waves travel at this velocity because the photon is massless. Now consider a relativistic train as shown in Fig. 3. There are 3 marks on the train and the track as shown. Point C is halfway between the points A and B. Observers on the train and the track are positioned at C. There is a lightening flash striking the train and the track at A and B at the same time. The observer on the tracks sees the light flash from A and B simultaneously and believes that the lightening struck the tracks at A and B at the same time. However the observer on the train moves to meet the flash from A and away from the flash at B. Thus this observer believes that A occured before B. The observers cannot agree on whether the events are simultaneous. In addition the observers cannot agree on whether the length AB or A B is longer, since this length is also measured by determining when the light from A and B reached the center of the train. Therefore, time and length depend on the frame of reference. We incorporate this into the coordinate transformation by writing; x = x 9

10 B C A Vo B C A Figure 3: Simultaneous events and a moving train y = y z = f(z, t) t = g(z, t) Here f, g are functions to be determined. To do this we assume; 3) Space time is isotropic and homogeneous. This means we need to choose linear functions so that any point in space is weighted the same as any other. 4) The relations reduce to the Galilean transformation in the limit of low velocities. Applying these assumptions the equations have the form, x = x y = y z = α(z V 0 t) t = rz + st Now use the fact that c is constant in all frames of reference. x 2 + y 2 + z 2 c 2 t 2 = x 2 + y 2 + z 2 c 2 t 2 Substitute the above transformations and collect terms in x, y, z, t. Because the variables are linearly independent, the coefficients of these variables must vanish. This results in the solution we seek - the Lorentz transformation. x = x 10

11 y = y z = γ(z V 0 t) t = γ(t (v 0 /c 2 )z) with γ = 1 1 β 2 and β = V 0 /c 7 Length transformation A length measurement occurs by comparing the position of the ends at the same time. Thus a length in the direction of motion is; L = (z 2 z 1 ) = γ(z 2 ct) γ(z 1 ct) Therefore; L = γl Since γ > 1 and we choose the primed system to be at rest, it means that the moving length appears shorter - a length contraction. 8 Time transformation To measure time we must compare the rate of two clocks. We arrange the clocks so that their times are synchronized to t = 0 at the position z = 0. At some later time the moving clock has traveled a distance d = V 0 t. The time difference between the clocks requires that the information travel back to the clock at rest from this distance. Then the time difference is; τ = γ(t 2 v 0 /c 2 z 2 ) 0 τ = γ(t 2 v 0 /c 2 V 0 t) τ = τ/γ 11

12 9 Example Muons are produced by the collision of cosmic ray protons with atoms in the upper atmosphere of the earth. Muons are unstable particles and decay with a meanlife of approximately s when at rest. These muons, however, are moving at relativistic speed after the collision. Consider two observers, one moving with the muons and one on the earth s surface. To the observer on the earth, the number of muons reaching the surface is given by the decay equation; N = N 0 e λt where N is the number reaching the surface, N 0 is the number produced at altitude, λ is the meanlife, and t is the time in the earth-observer s frame of reference. Since λ is the meanlife in the rest frame, the time in the earth frame must be changed to the time in the rest frame, t = γ t. The distance traveled is d at velocity V, so the time in the earth frame is t = d/v. Thus the number observed is; N = N 0 e λt /γ = N 0 e λdt /(γv ) On the other hand when traveling in the muon frame, the muon travels a distance that is contracted with respect to the distance in the earth frame due to the muon velocity, d = d/γ. The time to reach the surface is t = d/(γv). Thus in this frame the number observed at the surface is; N = N 0 e λt /γ = N 0 e λdt /(γv ) This is the same as in the earth frame but the interpretation differs. In the earth frame the time is dialated by the muon velocity. In the muon frame the distance is contracted. 12