text:
Ruh-Roh… Polls Overestimated Obama's Strength By Average of 7 Percentage Points This Year
Here's something the Obamatrons to chew on for the next 15 days…
Polls this year have overestimated Obama's strength in 18 of 20 states by an average of 7%.
Ace of Spades and The Politico reported:
In theory, exit polls should match election results. But for all the care that goes into conducting accurate exit polls, errant results aren't completely uncommon. Respected polling analyst Mark Blumenthal found that during the Democratic primaries this year, preliminary exit polls overestimated Obama's strength in 18 of 20 states, by an average error of 7 percentage points, based on leaked early results.
The reason? Obama's supporters were younger, better educated and often more enthusiastic than Hillary Clinton's, meaning they were more likely to participate in exit polls.
Jim Geraghty sees a tight race likely going to McCain-Palin.
Joshua also linked to this post by DJ Drummond at Stolen Thunder on bogus liberal polls.
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text: EUSTIS — A business expo held Tuesday at the Eustis Community Center drew 35 Lake County companies ranging from computer services to diesel equipment. The first Lake County Business Expo was organized by Rob Newton, 32, who operates a sales and marketing business from his Eustis home. Newton said the purpose of the show was for residents and business people to become familiar with products and services offered by companies in the county.


text:
Introduction to Tensor Calculus for General Relativity
- Augustine Simmons
- 3 years ago
- Views:
1 Massachusetts Institute of Technology Department of Physics Physics Spring 1999 Introduction to Tensor Calculus for General Relativity c 1999 Edmund Bertschinger. All rights reserved. 1 Introduction There are three essential ideas underlying general relativity (GR). The first is that spacetime may be described as a curved, four-dimensional mathematical structure called a pseudo-riemannian manifold. In brief, time and space together comprise a curved fourdimensional non-euclidean geometry. Consequently, the practitioner of GR must be familiar with the fundamental geometrical properties of curved spacetime. In particular, the laws of physics must be expressed in a form that is valid independently of any coordinate system used to label points in spacetime. The second essential idea underlying GR is that at every spacetime point there exist locally inertial reference frames, corresponding to locally flat coordinates carried by freely falling observers, in which the physics of GR is locally indistinguishable from that of special relativity. This is Einstein s famous strong equivalence principle and it makes general relativity an extension of special relativity to a curved spacetime. The third key idea is that mass (as well as mass and momentum flux) curves spacetime in a manner described by the tensor field equations of Einstein. These three ideas are exemplified by contrasting GR with Newtonian gravity. In the Newtonian view, gravity is a force accelerating particles through Euclidean space, while time is absolute. From the viewpoint of GR, there is no gravitational force. Rather, in the absence of electromagnetic and other forces, particles follow the straightest possible paths (geodesics) through a spacetime curved by mass. Freely falling particles define locally inertial reference frames. Time and space are not absolute but are combined into the four-dimensional manifold called spacetime. Working with GR, particularly with the Einstein field equations, requires some understanding of differential geometry. In these notes we will develop the essential mathematics needed to describe physics in curved spacetime. Many physicists receive their 1
2 introduction to this mathematics in the excellent book of Weinberg (1972). Weinberg minimizes the geometrical content of the equations by representing tensors using component notation. We believe that it is equally easy to work with a more geometrical description, with the additional benefit that geometrical notation makes it easier to distinguish physical results that are true in any coordinate system (e.g., those expressible using vectors) from those that are dependent on the coordinates. Because the geometry of spacetime is so intimately related to physics, we believe that it is better to highlight the geometry from the outset. In fact, using a geometrical approach allows us to develop the essential differential geometry as an extension of vector calculus. Our treatment is closer to that Wald (1984) and closer still to Misner, Thorne and Wheeler (1973). These books are rather advanced. For the newcomer to general relativity we warmly recommend Schutz (1985). Our notation and presentation is patterned largely after Schutz. The student wishing additional practice problems in GR should consult Lightman et al. (1975). A slightly more advanced mathematical treatment is provided in the excellent notes of Carroll (1997). These notes assume familiarity with special relativity. We will adopt units in which the speed of light c = 1. Greek indices (µ, ν, etc., which take the range {0, 1, 2, 3}) will be used to represent components of tensors. The Einstein summation convention is assumed: repeated upper and lower indices are to be summed over their ranges, e.g., A µ B µ A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3. Four-vectors will be represented with an arrow over the symbol, e.g., A, while one-forms will be represented using a tilde, e.g., B. Spacetime points will be denoted in boldface type; e.g., x refers to a point with coordinates x µ. Our metric has signature +2; the flat spacetime Minkowski metric components are η µν = diag( 1, +1, +1, +1). 2 Vectors and one-forms The essential mathematics of general relativity is differential geometry, the branch of mathematics dealing with smoothly curved surfaces (differentiable manifolds). The physicist does not need to master all of the subtleties of differential geometry in order to use general relativity. (For those readers who want a deeper exposure to differential geometry, see the introductory texts of Lovelock and Rund 1975, Bishop and Goldberg 1980, or Schutz 1980.) It is sufficient to develop the needed differential geometry as a straightforward extension of linear algebra and vector calculus. However, it is important to keep in mind the geometrical interpretation of physical quantities. For this reason, we will not shy from using abstract concepts like points, curves and vectors, and we will distinguish between a vector A and its components A µ. Unlike some other authors (e.g., Weinberg 1972), we will introduce geometrical objects in a coordinate-free manner, only later introducing coordinates for the purpose of simplifying calculations. This approach 2
3 requires that we distinguish vectors from the related objects called one-forms. Once the differences and similarities between vectors, one-forms and tensors are clear, we will adopt a unified notation that makes computations easy. 2.1 Vectors We begin with vectors. A vector is a quantity with a magnitude and a direction. This primitive concept, familiar from undergraduate physics and mathematics, applies equally in general relativity. An example of a vector is d x, the difference vector between two infinitesimally close points of spacetime. Vectors form a linear algebra (i.e., a vector space). If A is a vector and a is a real number (scalar) then a A is a vector with the same direction (or the opposite direction, if a < 0) whose length is multiplied by a. If A and B are vectors then so is A + B. These results are as valid for vectors in a curved four-dimensional spacetime as they are for vectors in three-dimensional Euclidean space. Note that we have introduced vectors without mentioning coordinates or coordinate transformations. Scalars and vectors are invariant under coordinate transformations; vector components are not. The whole point of writing the laws of physics (e.g., F = m a) using scalars and vectors is that these laws do not depend on the coordinate system imposed by the physicist. We denote a spacetime point using a boldface symbol: x. (This notation is not meant to imply coordinates.) Note that x refers to a point, not a vector. In a curved spacetime the concept of a radius vector x pointing from some origin to each point x is not useful because vectors defined at two different points cannot be added straightforwardly as they can in Euclidean space. For example, consider a sphere embedded in ordinary three-dimensional Euclidean space (i.e., a two-sphere). A vector pointing east at one point on the equator is seen to point radially outward at another point on the equator whose longitude is greater by 90. The radially outward direction is undefined on the sphere. Technically, we are discussing tangent vectors that lie in the tangent space of the manifold at each point. For example, a sphere may be embedded in a three-dimensional Euclidean space into which may be placed a plane tangent to the sphere at a point. A twodimensional vector space exists at the point of tangency. However, such an embedding is not required to define the tangent space of a manifold (Walk 1984). As long as the space is smooth (as assumed in the formal definition of a manifold), the difference vector d x between to infinitesimally close points may be defined. The set of all d x defines the tangent space at x. By assigning a tangent vector to every spacetime point, we can recover the usual concept of a vector field. However, without additional preparation one cannot compare vectors at different spacetime points, because they lie in different tangent spaces. In Section 5 we introduce parallel transport as a means of making this comparison. Until then, we consider only tangent vectors at x. To emphasize the status 3
4 of a tangent vector, we will occasionally use a subscript notation: AX. 2.2 One-forms and dual vector space Next we introduce one-forms. A one-form is defined as a linear scalar function of a vector. That is, a one-form takes a vector as input and outputs a scalar. For the one-form P, P( V ) is also called the scalar product and may be denoted using angle brackets: P( V ) = P, V. (1) The one-form is a linear function, meaning that for all scalars a and b and vectors V and W, the one-form P satisfies the following relations: P(a V + b W) = P, a V + b W = a P, V + b P, W = a P( V ) + b P( W). (2) Just as we may consider any function f( ) as a mathematical entity independently of any particular argument, we may consider the one-form P independently of any particular vector V. We may also associate a one-form with each spacetime point, resulting in a one-form field P = P X. Now the distinction between a point a vector is crucial: PX is a one-form at point x while P( V ) is a scalar, defined implicitly at point x. The scalar product notation with subscripts makes this more clear: P X, V X. One-forms obey their own linear algebra distinct from that of vectors. Given any two scalars a and b and one-forms P and Q, we may define the one-form a P + b Q by (a P + b Q)( V ) = a P + b Q, V = a P, V + b Q, V = a P( V ) + b Q( V ). (3) Comparing equations (2) and (3), we see that vectors and one-forms are linear operators on each other, producing scalars. It is often helpful to consider a vector as being a linear scalar function of a one-form. Thus, we may write P, V = P( V ) = V ( P). The set of all one-forms is a vector space distinct from, but complementary to, the linear vector space of vectors. The vector space of one-forms is called the dual vector (or cotangent) space to distinguish it from the linear space of vectors (tangent space). Although one-forms may appear to be highly abstract, the concept of dual vector spaces is familiar to any student of quantum mechanics who has seen the Dirac bra-ket notation. Recall that the fundamental object in quantum mechanics is the state vector, represented by a ket ψ in a linear vector space (Hilbert space). A distinct Hilbert space is given by the set of bra vectors φ. Bra vectors and ket vectors are linear scalar functions of each other. The scalar product φ ψ maps a bra vector and a ket vector to a scalar called a probability amplitude. The distinction between bras and kets is necessary because probability amplitudes are complex numbers. As we will see, the distinction between vectors and one-forms is necessary because spacetime is curved. 4
5 3 Tensors Having defined vectors and one-forms we can now define tensors. A tensor of rank (m, n), also called a (m, n) tensor, is defined to be a scalar function of m one-forms and n vectors that is linear in all of its arguments. It follows at once that scalars are tensors of rank (0, 0), vectors are tensors of rank (1, 0) and one-forms are tensors of rank (0, 1). We may denote a tensor of rank (2, 0) by T( P, Q); one of rank (2, 1) by T( P, Q, A), etc. Our notation will not distinguish a (2, 0) tensor T from a (2, 1) tensor T, although a notational distinction could be made by placing m arrows and n tildes over the symbol, or by appropriate use of dummy indices (Wald 1984). The scalar product is a tensor of rank (1, 1), which we will denote I and call the identity tensor: I( P, V ) P, V = P( V ) = V ( P). (4) We call I the identity because, when applied to a fixed one-form P and any vector V, it returns P( V ). Although the identity tensor was defined as a mapping from a one-form and a vector to a scalar, we see that it may equally be interpreted as a mapping from a one-form to the same one-form: I( P, ) = P, where the dot indicates that an argument (a vector) is needed to give a scalar. A similar argument shows that I may be considered the identity operator on the space of vectors V : I(, V ) = V. A tensor of rank (m, n) is linear in all its arguments. For example, for (m = 2, n = 0) we have a straightforward extension of equation (2): T(a P + b Q, c R + d S) = ac T( P, R) + ad T( P, S) + bc T( Q, R) + bd T( q, S). (5) Tensors of a given rank form a linear algebra, meaning that a linear combination of tensors of rank (m, n) is also a tensor of rank (m, n), defined by straightforward extension of equation (3). Two tensors (of the same rank) are equal if and only if they return the same scalar when applied to all possible input vectors and one-forms. Tensors of different rank cannot be added or compared, so it is important to keep track of the rank of each tensor. Just as in the case of scalars, vectors and one-forms, tensor fields TX are defined by associating a tensor with each spacetime point. There are three ways to change the rank of a tensor. The first, called the tensor (or outer) product, combines two tensors of ranks (m 1, n 1 ) and (m 2, n 2 ) to form a tensor of rank (m 1 + m 2, n 1 + n 2 ) by simply combining the argument lists of the two tensors and thereby expanding the dimensionality of the tensor space. For example, the tensor product of two vectors A and B gives a rank (2, 0) tensor T = A B, T( P, Q) A( P) B( Q). (6) We use the symbol to denote the tensor product; later we will drop this symbol for notational convenience when it is clear from the context that a tensor product is implied. 5
6 Note that the tensor product is non-commutative: A B B A (unless B = c A for some scalar c) because A( P) B( Q) A( Q) B( P) for all P and Q. We use the symbol to denote the tensor product of any two tensors, e.g., P T = P A B is a tensor of rank (2, 1). The second way to change the rank of a tensor is by contraction, which reduces the rank of a (m, n) tensor to (m 1, n 1). The third way is the gradient. We will discuss contraction and gradients later. 3.1 Metric tensor The scalar product (eq. 1) requires a vector and a one-form. Is it possible to obtain a scalar from two vectors or two one-forms? From the definition of tensors, the answer is clearly yes. Any tensor of rank (0, 2) will give a scalar from two vectors and any tensor of rank (2, 0) combines two one-forms to give a scalar. However, there is a particular (0, 2) tensor field g X called the metric tensor and a related (2, 0) tensor field g 1 X called the inverse metric tensor for which special distinction is reserved. The metric tensor is a symmetric bilinear scalar function of two vectors. That is, given vectors V and W, g returns a scalar, called the dot product: g( V, W) = V W = W V = g( W, V ). (7) Similarly, g 1 returns a scalar from two one-forms P and Q, which we also call the dot product: g 1 ( P, Q) = P Q = P Q = g 1 ( P, Q). (8) Although a dot is used in both cases, it should be clear from the context whether g or g 1 is implied. We reserve the dot product notation for the metric and inverse metric tensors just as we reserve the angle brackets scalar product notation for the identity tensor (eq. 4). Later (in eq. 40) we will see what distinguishes g from other (0, 2) tensors and g 1 from other symmetric (2, 0) tensors. One of the most important properties of the metric is that it allows us to convert vectors to one-forms. If we forget to include W in equation (7) we get a quantity, denoted Ṽ, that behaves like a one-form: Ṽ ( ) g( V, ) = g(, V ), (9) where we have inserted a dot to remind ourselves that a vector must be inserted to give a scalar. (Recall that a one-form is a scalar function of a vector!) We use the same letter to indicate the relation of V and Ṽ. Thus, the metric g is a mapping from the space of vectors to the space of one-forms: g : V Ṽ. By definition, the inverse metric g 1 is the inverse mapping: g 1 : Ṽ V. 6
7 (The inverse always exists for nonsingular spacetimes.) Thus, if Ṽ is defined for any V by equation (9), the inverse metric tensor is defined by V ( ) g 1 (Ṽ, ) = g 1 (, Ṽ ). (10) Equations (4) and (7) (10) give us several equivalent ways to obtain scalars from vectors V and W and their associated one-forms Ṽ and W: Ṽ, W = W, V = V W = Ṽ W = I(Ṽ, W) = I( W, V ) = g( V, W) = g 1 (Ṽ, W). (11) 3.2 Basis vectors and one-forms It is possible to formulate the mathematics of general relativity entirely using the abstract formalism of vectors, forms and tensors. However, while the geometrical (coordinate-free) interpretation of quantities should always be kept in mind, the abstract approach often is not the most practical way to perform calculations. To simplify calculations it is helpful to introduce a set of linearly independent basis vector and one-form fields spanning our vector and dual vector spaces. In the same way, practical calculations in quantum mechanics often start by expanding the ket vector in a set of basis kets, e.g., energy eigenstates. By definition, the dimensionality of spacetime (four) equals the number of linearly independent basis vectors and one-forms. We denote our set of basis vector fields by { e µx }, where µ labels the basis vector (e.g., µ = 0, 1, 2, 3) and x labels the spacetime point.. Any four linearly independent basis vectors at each spacetime point will work; we do not not impose orthonormality or any other conditions in general, nor have we implied any relation to coordinates (although later we will). Given a basis, we may expand any vector field A as a linear combination of basis vectors: A X = A µ X e µx = A 0 X e 0X + A 1 X e 1X + A 2 X e 2X + A 3 X e 3X. (12) Note our placement of subscripts and superscripts, chosen for consistency with the Einstein summation convention, which requires pairing one subscript with one superscript. The coefficients A µ are called the components of the vector (often, the contravariant components). Note well that the coefficients A µ depend on the basis vectors but A does not! Similarly, we may choose a basis of one-form fields in which to expand one-forms like ÃX. Although any set of four linearly independent one-forms will suffice for each spacetime point, we prefer to choose a special one-form basis called the dual basis and denoted {ẽ µ X}. Note that the placement of subscripts and superscripts is significant; we never use a subscript to label a basis one-form while we never use a superscript to label a basis vector. Therefore, ẽ µ is not related to e µ through the metric (eq. 9): 7
8 ẽ µ ( ) g( e µ, ). Rather, the dual basis one-forms are defined by imposing the following 16 requirements at each spacetime point: ẽ µ X, e ν X = δ µ ν, (13) where δ µ ν is the Kronecker delta, δµ ν = 1 if µ = ν and δµ ν = 0 otherwise, with the same values for each spacetime point. (We must always distinguish subscripts from superscripts; the Kronecker delta always has one of each.) Equation (13) is a system of four linear equations at each spacetime point for each of the four quantities ẽ µ and it has a unique solution. (The reader may show that any nontrivial transformation of the dual basis one-forms will violate eq. 13.) Now we may expand any one-form field P X in the basis of one-forms: P X = P µx ẽ µ X. (14) The component P µ of the one-form P is often called the covariant component to distinguish it from the contravariant component P µ of the vector P. In fact, because we have consistently treated vectors and one-forms as distinct, we should not think of these as being distinct components of the same entity at all. There is a simple way to get the components of vectors and one-forms, using the fact that vectors are scalar functions of one-forms and vice versa. One simply evaluates the vector using the appropriate basis one-form: and conversely for a one-form: A(ẽ µ ) = ẽ µ, A = ẽ µ, A ν e ν = ẽ µ, e ν A ν = δ µ νa ν = A µ, (15) P( e µ ) = P, e µ = P ν ẽ ν, e µ = ẽ ν, e µ P ν = δ ν µp ν = P µ. (16) We have suppressed the spacetime point x for clarity, but it is always implied. 3.3 Tensor algebra We can use the same ideas to expand tensors as products of components and basis tensors. First we note that a basis for a tensor of rank (m, n) is provided by the tensor product of m vectors and n one-forms. For example, a (0, 2) tensor like the metric tensor can be decomposed into basis tensors ẽ µ ẽ ν. The components of a tensor of rank (m, n), labeled with m superscripts and n subscripts, are obtained by evaluating the tensor using m basis one-forms and n basis vectors. For example, the components of the (0, 2) metric tensor, the (2, 0) inverse metric tensor and the (1, 1) identity tensor are g µν g( e µ, e ν ) = e µ e ν, g µν g 1 (ẽ µ, ẽ ν ) = ẽ µ ẽ ν, δ µ ν = I(ẽ µ, e ν ) = ẽ µ, e ν. (17) 8
9 (The last equation follows from eqs 4 and 13.) The tensors are given by summing over the tensor product of basis vectors and one-forms: g = g µν ẽ µ ẽ ν, g 1 = g µν e µ e ν, I = δ µ ν e µ ẽ ν. (18) The reader should check that equation (18) follows from equations (17) and the duality condition equation (13). Basis vectors and one-forms allow us to represent any tensor equations using components. For example, the dot product between two vectors or two one-forms and the scalar product between a one-form and a vector may be written using components as A B = g µν A µ A ν, P, A = P µ A µ, P Q = g µν P µ P ν. (19) The reader should prove these important results. If two tensors of the same rank are equal in one basis, i.e., if all of their components are equal, then they are equal in any basis. While this mathematical result is obvious from the basis-free meaning of a tensor, it will have important physical implications in GR arising from the Equivalence Principle. As we discussed above, the metric and inverse metric tensors allow us to transform vectors into one-forms and vice versa. If we evaluate the components of V and the one-form Ṽ defined by equations (9) and (10), we get V µ = g( e µ, V ) = g µν V ν, V µ = g 1 (ẽ µ, Ṽ ) = gµν V ν. (20) Because these two equations must hold for any vector V, we conclude that the matrix defined by g µν is the inverse of the matrix defined by g µν : g µκ g κν = δ µ ν. (21) We also see that the metric and its inverse are used to lower and raise indices on components. Thus, given two vectors V and W, we may evaluate the dot product any of four equivalent ways (cf. eq. 11): V W = g µν V µ W ν = V µ W µ = V µ W µ = g µν V µ W ν. (22) In fact, the metric and its inverse may be used to transform tensors of rank (m, n) into tensors of any rank (m + k, n k) where k = m, m + 1,...,n. Consider, for example, a (1, 2) tensor T with components T µ νλ T(ẽµ, e ν, e λ ). (23) If we fail to plug in the one-form ẽ µ, the result is the vector T κ νλ e κ. (A one-form must be inserted to return the quantity T κ νλ.) This vector may then be inserted into the metric tensor to give the components of a (0, 3) tensor: T µνλ g( e µ, T κ νλ e κ ) = g µκ T κ νλ. (24) 9
10 We could now use the inverse metric to raise the third index, say, giving us the component of a (1, 2) tensor distinct from equation (23): T λ µν g 1 (ẽ λ, T µνκ ẽ κ ) = g λκ T µνκ = g λκ g µρ T ρ νκ. (25) In fact, there are 2 m+n different tensor spaces with ranks summing to m+n. The metric or inverse metric tensor allow all of these tensors to be transformed into each other. Returning to equation (22), we see why we must distinguish vectors (with components V µ ) from one-forms (with components V µ ). The scalar product of two vectors requires the metric tensor while that of two one-forms requires the inverse metric tensor. In general, g µν g µν. The only case in which the distinction is unnecessary is in flat (Lorentz) spacetime with orthonormal Cartesian basis vectors, in which case g µν = η µν is everywhere the diagonal matrix with entries ( 1, +1, +1, +1). However, gravity curves spacetime. (Besides, we may wish to use curvilinear coordinates even in flat spacetime.) As a result, it is impossible to define a coordinate system for which g µν = g µν everywhere. We must therefore distinguish vectors from one-forms and we must be careful about the placement of subscripts and superscripts on components. At this stage it is useful to introduce a classification of vectors and one-forms drawn from special relativity with its Minkowski metric η µν. Recall that a vector A = A µ e µ is called spacelike, timelike or null according to whether A A = η µν A µ A ν is positive, negative or zero, respectively. In a Euclidean space, with positive definite metric, A A is never negative. However, in the Lorentzian spacetime geometry of special relativity, time enters the metric with opposite sign so that it is possible to have A A < 0. In particular, the four-velocity u µ = dx µ /dτ of a massive particle (where dτ is proper time) is a timelike vector. This is seen most simply by performing a Lorentz boost to the rest frame of the particle in which case u t = 1, u x = u y = u z = 0 and η µν u µ u ν = 1. Now, η µν u µ u ν is a Lorentz scalar so that u u = 1 in any Lorentz frame. Often this is written p p = m 2 where p µ = mu µ is the four-momentum for a particle of mass m. For a massless particle (e.g., a photon) the proper time vanishes but the four-momentum is still well-defined with p p = 0: the momentum vector is null. We adopt the same notation in general relativity, replacing the Minkowski metric (components η µν ) with the actual metric g and evaluating the dot product using A A = g( A, A) = g µν A µ A ν. The same classification scheme extends to one-forms using g 1 : a one-form P is spacelike, timelike or null according to whether P P = g 1 ( P, P) = g µν P µ P ν is positive, negative or zero, respectively. Finally, a vector is called a unit vector if A A = ±1 and similarly for a one-form. The four-velocity of a massive particle is a timelike unit vector. Now that we have introduced basis vectors and one-forms, we can define the contraction of a tensor. Contraction pairs two arguments of a rank (m, n) tensor: one vector and one one-form. The arguments are replaced by basis vectors and one-forms and summed over. For example, consider the (1, 3) tensor R, which may be contracted on its second 10
11 vector argument to give a (0, 2) tensor also denoted R but distinguished by its shorter argument list: R( A, B) = δ λ κ R(ẽ κ, A, e λ, B) = 3 λ=0 R(ẽ λ, A, e λ, B). (26) In Section 6 we will define the Riemann curvature tensor of rank (1, 3); its contraction, defined by equation (26), is called the Ricci tensor. Although the contracted tensor would appear to depend on the choice of basis because its definition involves the basis vectors and one-forms, the reader may show that it is actually invariant under a change of basis (and is therefore a tensor) as long as we use dual one-form and vector bases satisfying equation (13). Equation (26) becomes somewhat clearer if we express it entirely using tensor components: R µν = R λ µλν. (27) Summation over λ is implied. Contraction may be performed on any pair of covariant and contravariant indices; different tensors result. 3.4 Change of basis We have made no restrictions upon our choice of basis vectors e µ. Our basis vectors are simply a linearly independent set of four vector fields allowing us to express any vector as a linear combination of basis vectors. The choice of basis is not unique; we may transform from one basis to another simply by defining four linearly independent combinations of our original basis vectors. Given one basis { e µx }, we may define another basis { e µ X}, distinguished by placing a prime on the labels, as follows: e µ X = Λ ν µ X e ν X. (28) The prime is placed on the index rather than on the vector only for notational convenience; we do not imply that the new basis vectors are a permutation of the old ones. Any linearly independent linear combination of old basis vectors may be selected as the new basis vectors. That is, any nonsingular four-by-four matrix may be used for Λ ν µ. The transformation is not restricted to being a Lorentz transformation; the local reference frames defined by the bases are not required to be inertial (or, in the presence of gravity, freely-falling). Because the transformation matrix is assumed to be nonsingular, the transformation may be inverted: e ν X = Λ µ ν X e µ X, Λ µ κ Λκ ν δµ ν. (29) Comparing equations (28) and (29), note that in our notation the inverse matrix places the prime on the other index. Primed indices are never summed together with unprimed indices. 11
12 If we change basis vectors, we must also transform the basis one-forms so as to preserve the duality condition equation (13). The reader may verify that, given the transformations of equations (28) and (29), the new dual basis one-forms are ẽ µ X = Λ µ ν Xẽ ν X. (30) We may also write the transformation matrix and its inverse by scalar products of the old and new basis vectors and one-forms (dropping the subscript x for clarity): Λ ν µ = ẽν, e µ, Λ µ ν = ẽµ, e ν. (31) Apart from the basis vectors and one-forms, a vector A and a one-form P are, by definition, invariant under a change of basis. Their components are not. For example, using equation (29) or (31) we find A = A ν e ν = A µ e µ, A µ = ẽ µ, A = Λ µ νa ν. (32) The vector components transform oppositely to the basis vectors (eq. 28). One-form components transform like basis vectors, as suggested by the fact that both are labeled with a subscript: Ã = A ν ẽ ν = A µ ẽ µ, A µ = Ã, e µ = Λν µ A ν. (33) Note that if the components of two vectors or two one-forms are equal in one basis, they are equal in any basis. Tensor components also transform under a change of basis. The new components may be found by recalling that a (m, n) tensor is a function of m vectors and n one-forms and that its components are gotten by evaluating the tensor using the basis vectors and one-forms (e.g., eq. 17). For example, the metric components are transformed under the change of basis of equation (28) to g µ ν g( e µ, e ν ) = g αβ ẽ α ( e µ )ẽ β ( e ν ) = g αβ Λ α µ Λβ ν. (34) (Recall that evaluating a one-form or vector means using the scalar product, eq. 1.) We see that the covariant components of the metric (i.e., the lower indices) transform exactly like one-form components. Not surprisingly, the components of a tensor of rank (m, n) transform like the product of m vector components and n one-form components. If the components of two tensors of the same rank are equal in one basis, they are equal in any basis. 12
13 3.5 Coordinate bases We have made no restrictions upon our choice of basis vectors e µ. Before concluding our formal introduction to tensors, we introduce one more idea: a coordinate system. A coordinate system is simply a set of four differentiable scalar fields x µ X (not one vector field note that µ labels the coordinates and not vector components) that attach a unique set of labels to each spacetime point x. That is, no two points are allowed to have identical values of all four scalar fields and the coordinates must vary smoothly throughout spacetime (although we will tolerate occasional flaws like the coordinate singularities at r = 0 and θ = 0 in spherical polar coordinates). Note that we impose no other restrictions on the coordinates. The freedom to choose different coordinate systems is available to us even in a Euclidean space; there is nothing sacred about Cartesian coordinates. This is even more true in a non-euclidean space, where Cartesian coordinates covering the whole space do not exist. Coordinate systems are useful for three reasons. First and most obvious, they allow us to label each spacetime point by a set of numbers (x 0, x 1, x 2, x 3 ). The second and more important use is in providing a special set of basis vectors called a coordinate basis. Suppose that two infinitesimally close spacetime points have coordinates x µ and x µ + dx µ. The infinitesimal difference vector between the two points, denoted d x, is a vector defined at x. We define the coordinate basis as the set of four basis vectors e µx such that the components of d x are dx µ : d x dx µ e µ defines e µ in a coordinate basis. (35) From the trivial appearance of this equation the reader may incorrectly think that we have imposed no constraints on the basis vectors. However, that is not so. According to equation (35), the basis vector e 0X, for example, must point in the direction of increasing x 0 at point x. This corresponds to a unique direction in four-dimensional spacetime just as the direction of increasing latitude corresponds to a unique direction (north) at a given point on the earth. In more mathematical treatments (e.g. Walk 1984), e µ is associated with the directional derivative / x µ at x. Note that not all bases are coordinate bases. If we wanted to be perverse we could define a non-coordinate basis by, for example, permuting the labels on the basis vectors but not those on the coordinates (which, after all, are not the components of a vector). In this case ẽ µ, d x, the component of d x for basis vector e µ, would not equal the coordinate differential dx µ. This would violate nothing we have written so far except equation (35). Later we will discover more natural ways that non-coordinate bases may arise. The coordinate basis { e µ } defined by equation (35) has a dual basis of one-forms {ẽ µ } defined by equation (13). The dual basis of one-forms is related to the gradient. We obtain this relation as follows. Consider any scalar field f X. Treating f as a function of 13
14 the coordinates, the difference in f between two infinitesimally close points is df = f x µ dxµ µ f dx µ. (36) Equation (36) may be taken as the definition of the components of the gradient (with an alternative brief notation for the partial derivative). However, partial derivatives depend on the coordinates, while the gradient (covariant derivative) should not. What, then, is the gradient is it a vector or a one-form? From equation (36), because df is a scalar and dx µ is a vector component, f/ x µ must be the component of a one-form, not a vector. The notation µ, with its covariant (subscript) index, reinforces our view that the partial derivative is the component of a one-form and not a vector. We denote the gradient one-form by. Like all one-forms, The gradient may be decomposed into a sum over basis one-forms ẽ µ. Using equation (36) and the equation (13) as the requirement for a dual basis, we conclude that the gradient is ẽ µ µ in a coordinate basis. (37) Note that we must write the basis one-form to the left of the partial derivative operator, for the basis one-form itself may depend on position! We will return to this point in Section 4 when we discuss the covariant derivative. In the present case, it is clear from equation (36) that we must let the derivative act only on the function f. We can now rewrite equation (36) in the coordinate-free manner df = f, d x. (38) If we want the directional derivative of f along any particular direction, we simply replace d x by a vector pointing in the desired direction (e.g., the tangent vector to some curve). Also, if we let f X equal one of the coordinates, using equation (37) the gradient gives us the corresponding basis one-form: x µ = ẽ µ in a coordinate basis. (39) The third use of coordinates is that they can be used to describe the distance between two points of spacetime. However, coordinates alone are not enough. We also need the metric tensor. We write the squared distance between two spacetime points as ds 2 = d x 2 g(d x, d x) = d x d x. (40) IThis equation, true in any basis because it is a scalar equation that makes no reference to components, is taken as the definition of the metric tensor. Up to now the metric could have been any symmetric (0, 2) tensor. But, if we insist on being able to measure distances, given an infinitesimal difference vector d x, only one (0, 2) tensor can give the 14
15 squared distance. We define the metric tensor to be that tensor. Indeed, the squared magnitude of any vector A is A 2 g( A, A). Now we specialize to a coordinate basis, using equation (35) for d x. In a coordinate basis (and only in a coordinate basis), the squared distance is called the line element and takes the form ds 2 = g µν X dx µ dx ν in a coordinate basis. (41) We have used equation (17) to get the metric components. If we transform coordinates, we will have to change our vector and one-form bases. Suppose that we transform from x µ X to xµ X, with a prime indicating the new coordinates. For example, in the Euclidean plane we could transform from Cartesian coordinate (x 1 = x, x 2 = y) to polar coordinates (x 1 = r, x 2 = θ): x = r cos θ, y = r sin θ. A one-to-one mapping is given from the old to new coordinates, allowing us to define the Jacobian matrix Λ µ ν / x ν and its inverse Λ ν xµ µ = xν / x µ. Vector components transform like dx µ = ( x µ / x ν ) dx ν. Transforming the basis vectors, basis one-forms, and tensor components is straightforward using equations (28) (34). The reader should verify that equations (35), (37), (39) and (41) remain valid after a coordinate transformation. We have now introduced many of the basic ingredients of tensor algebra that we will need in general relativity. Before moving on to more advanced concepts, let us reflect on our treatment of vectors, one-forms and tensors. The mathematics and notation, while straightforward, are complicated. Can we simplify the notation without sacrificing rigor? One way to modify our notation would be to abandon ths basis vectors and one-forms and to work only with components of tensors. We could have defined vectors, one-forms and tensors from the outset in terms of the transformation properties of their components. However, the reader should appreciate the clarity of the geometrical approach that we have adopted. Our notation has forced us to distinguish physical objects like vectors from basis-dependent ones like vector components. As long as the definition of a tensor is not forgotten, computations are straightforward and unambiguous. Moreover, adopting a basis did not force us to abandon geometrical concepts. On the contrary, computations are made easier and clearer by retaining the notation and meaning of basis vectors and one-forms. 3.6 Isomorphism of vectors and one-forms Although vectors and one-forms are distinct objects, there is a strong relationship between them. In fact, the linear space of vectors is isomorphic to the dual vector space of one-forms (Wald 1984). Every equation or operation in one space has an equivalent equation or operation in the other space. This isomorphism can be used to hide the distinction between one-forms and vectors in a way that simplifies the notation. This approach is unusual (I haven t seen it published anywhere) and is not recommended in 15
16 formal work but it may be pedagogically useful. As we saw in equations (9) and (10), the link between the vector and dual vector spaces is provided by g and g 1. If A = B (components A µ = B µ ), then à = B (components A µ = B µ ) where A µ = g µν A ν and B µ = g µν B ν. So, why do we bother with one-forms when vectors are sufficient? The answer is that tensors may be functions of both one-forms and vectors. However, there is also an isomorphism among tensors of different rank. We have just argued that the tensor spaces of rank (1, 0) (vectors) and (0, 1) are isomorphic. In fact, all 2 m+n tensor spaces of rank (m, n) with fixed m + n are isomorphic. The metric and inverse metric tensors link together these spaces, as exemplified by equations (24) and (25). The isomorphism of different tensor spaces allows us to introduce a notation that unifies them. We could effect such a unification by discarding basis vectors and one-forms and working only with components, using the components of the metric tensor and its inverse to relate components of different types of tensors as in equations (24) and (25). However, this would require sacrificing the coordinate-free geometrical interpretation of vectors. Instead, we introduce a notation that replaces one-forms with vectors and (m, n) tensors with (m + n, 0) tensors in general. We do this by replacing the basis one-forms ẽ µ with a set of vectors defined as in equation (10): e µ ( ) g 1 (ẽ µ, ) = g µν e ν ( ). (42) We will refer to e µ as a dual basis vector to contrast it from both the basis vector e µ and the basis one-form ẽ µ. The dots are present in equation (42) to remind us that a one-form may be inserted to give a scalar. However, we no longer need to use one-forms. Using equation (42), given the components A µ of any one-form Ã, we may form the vector A defined by equation (10) as follows: A = A µ e µ = A µ g µν e ν = A µ e µ. (43) The reader should verify that A = A µ e µ is invariant under a change of basis because e µ transforms like a basis one-form. The isomorphism of one-forms and vectors means that we can replace all one-forms with vectors in any tensor equation. Tildes may be replaced with arrows. The scalar product between a one-form and a vector is replaced by the dot product using the metric (eq. 10 or 42). The only rule is that we must treat a dual basis vector with an upper index like a basis one-form: e µ e ν = g µν, e µ e ν = ẽ µ, e ν = δ µ ν, e µ e ν = ẽ µ ẽ ν = g µν. (44) The reader should verify equations (44) using equations (17) and (42). Now, if we need the contravariant component of a vector, we can get it from the dot product with the 16
17 dual basis vector instead of from the scalar product with the basis one-form: A µ = e µ A = ẽ µ, A. (45) We may also apply this recipe to convert the gradient one-form (eq. 37) to a vector, though we must not allow the dual basis vector to be differentiated: = e µ µ = g µν e µ ν in a coordinate basis. (46) It follows at once that the dual basis vector (in a coordinate basis) is the vector gradient of the coordinate: e µ = x µ. This equation is isomorphic to equation (39). The basis vectors and dual basis vectors, through their tensor products, also give a basis for higher-rank tensors. Again, the rule is to replace the basis one-forms with the corresponding dual basis vectors. Thus, for example, we may write the rank (2, 0) metric tensor in any of four ways: g = g µν e µ e ν = g µ ν e µ e ν = g ν µ e µ e ν = g µν e µ e ν. (47) In fact, by comparing this with equation (18) the reader will see that what we have written is actually the inverse metric tensor g 1, which is isomorphic to g through the replacement of ẽ µ with e µ. But, what are the mixed components of the metric, g µ ν and gµ ν? From equations (13) and (42), we see that they both equal the Kronecker delta δ µ ν. Consequently, the metric tensor is isomorphic to the identity tensor as well as to its inverse! However, this is no miracle; it was guaranteed by our definition of the dual basis vectors and by the fact we defined g 1 to invert the mapping from vectors to one-forms implied by g. The reader may fear that we have defined away the metric by showing it to be isomorphic to the identity tensor. However, this is not the case. We need the metric tensor components to obtain e µ from e µ or A µ from A µ. We cannot take advantage of the isomorphism of different tensor spaces without the metric. Moreover, as we showed in equation (40), the metric plays a fundamental role in giving the squared magnitude of a vector. In fact, as we will see later, the metric contains all of the information about the geometrical properties of spacetime. Clearly, the metric must play a fundamental role in general relativity. 3.7 Example: Euclidean plane We close this section by applying tensor concepts to a simple example: the Euclidean plane. This flat two-dimensional space can be covered by Cartesian coordinates (x, y) with line element and metric components ds 2 = dx 2 + dy 2 g xx = g yy = 1, g xy = g yx = 0. (48) 17
18 We prefer to use the coordinate names themselves as component labels rather than using numbers (e.g. g xx rather than g 11 ). The basis vectors are denoted e x and e y, and their use in plane geometry and linear algebra is standard. Basis one-forms appear unnecessary because the metric tensor is just the identity tensor in this basis. Consequently the dual basis vectors (eq. 42) are e x = e x, e y = e y and no distinction is needed between superscripts and subscripts. However, there is nothing sacred about Cartesian coordinates. Consider polar coordinates (ρ, θ), defined by the transformation x = ρ cosθ, y = ρ sin θ. A simple exercise in partial derivatives yields the line element in polar coordinates: ds 2 = dρ 2 + ρ 2 dθ 2 g ρρ = 1, g θθ = ρ 2, g ρθ = g θρ = 0. (49) This appears eminently reasonable until, perhaps, one considers the basis vectors e ρ and e θ, recalling that g µν = e µ e ν. Then, while e ρ e ρ = 1 and e ρ e θ = 0, e θ e θ = ρ 2 : e θ is not a unit vector! The new basis vectors are easily found in terms of the Cartesian basis vectors and components using equation (28): e ρ = x x2 + y e y 2 x + x2 + y e 2 y, e θ = y e x + x e y. (50) The polar unit vectors are ˆρ = e ρ and ˆθ = ρ 1 e θ. Why does our formalism give us non-unit vectors? The answer is because we insisted that our basis vectors be a coordinate basis (eqs. 35, 37, 39 and 41). In terms of the orthonormal unit vectors, the difference vector between points (ρ, θ) and (ρ+dρ, θ+dθ) is d x = ˆρ dρ+ˆθ ρdθ. In the coordinate basis this takes the simpler form d x = e ρ dρ+ e θ dθ = dx µ e µ. In the coordinate basis we don t have to worry about normalizing our vectors; all information about lengths is carried instead by the metric. In the non-coordinate basis of orthonormal vectors {ˆρ, ˆθ} we have to make a separate note that the distance elements are dρ and ρdθ. In the non-coordinate basis we can no longer use equation (41) for the line element. We must instead use equation (40). The metric components in the non-coordinate basis {ˆρ, ˆθ} are gˆρˆρ = gˆθˆθ = 1, gˆρˆθ = gˆθˆρ = 0. (51) The reader may also verify this result by transforming the components of the metric from the basis { e ρ, e θ } to {ˆρ, ˆθ} using equation (34) with Λ ρ ˆρ = 1, Λθ ˆθ = ρ 1. Now, equation (40) still gives the distance squared, but we are responsible for remembering d x = ˆρ dρ + ˆθ ρdθ. In a non-coordinate basis, the metric will not tell us how to measure distances in terms of coordinate differentials. With a non-coordinate basis, we must sacrifice equations (35) and (41). Nonetheless, for some applications it proves convenient to introduce an orthonormal non-coordinate basis called a tetrad basis. Tetrads are discussed by Wald (1984) and Misner et al (1973). 18
19 The use of non-coordinate bases also complicates the gradient (eqs. 37, 39 and 46). In our polar coordinate basis (eq. 49), the inverse metric components are g ρρ = 1, g θθ = ρ 2, g ρθ = g θρ = 0. (52) (The matrix g µν is diagonal, so its inverse is also diagonal with entries given by the reciprocals.) The basis one-forms obey the rules ẽ µ ẽ ν = g µν. They are isomorphic to the dual basis vectors e µ = g µν e ν (eq. 42). Thus, e ρ = e ρ = ˆρ, e θ = ρ 2 e θ = ρ 1ˆθ. Equation (37) gives the gradient one-form as = ẽ ρ ( / ρ)+ẽ θ ( / θ). Expressing this as a vector (eq. 46) we get = e ρ ρ + e θ θ = ˆρ ρ + ˆθ 1 ρ θ. (53) The gradient is simpler in the coordinate basis. The coordinate basis has the added advantage that we can get the dual basis vectors (or the basis one-forms) by applying the gradient to the coordinates (eq. 46): e ρ = ρ, e θ = θ. From now on, unless otherwise noted, we will assume that our basis vectors are a coordinate basis. We will use one-forms and vectors interchangeably through the mapping provided by the metric and inverse metric (eqs. 9, 10 and 42). Readers who dislike one-forms may convert the tildes to arrows and use equations (44) to obtain scalars from scalar products and dot products. 4 Differentiation and Integration In this section we discuss differentiation and integration in curved spacetime. These might seem like a delicate subjects but, given the tensor algebra that we have developed, tensor calculus is straightforward. 4.1 Gradient of a scalar Consider first the gradient of a scalar field f X. We have already shown in Section 2 that the gradient operator is a one-form (an object that is invariant under coordinate transformations) and that, in a coordinate basis, its components are simply the partial derivatives with respect to the coordinates: f = ( µ f) ẽ µ = ( µ f) ẽ µ, (54) where µ ( / x µ ). We have introduced a second notation, µ, called the covariant derivative with respect to x µ. By definition, the covariant derivative behaves like the 19
20 component of a one-form. But, from equation (54), this is also true of the partial derivative operator µ. Why have we introduced a new symbol? Before answering this question, let us first note that the gradient, because it behaves like a tensor of rank (0, 1) (a one-form), changes the rank of a tensor field from (m, n) to (m, n + 1). (This is obviously true for the gradient of a scalar field, with m = n = 0.) That is, application of the gradient is like taking the tensor product with a one-form. The difference is that the components are not the product of the components, because µ is not a number. Nevertheless, the resulting object must be a tensor of rank (m, n + 1); e.g., its components must transform like components of a (m, n+1) tensor. The gradient of a scalar field f is a (0, 1) tensor with components ( µ f). 4.2 Gradient of a vector: covariant derivative The reason that we have introduced a new symbol for the derivative will become clear when we take the gradient of a vector field A X = A µ X e µx. In general, the basis vectors are functions of position as are the vector components! So, the gradient must act on both. In a coordinate basis, we have A = (A ν e ν ) = ẽ µ µ (A ν e ν ) = ( µ A ν ) ẽ µ e ν + A ν ẽ µ ( µ e ν ) ( µ A ν ) ẽ µ e ν. (55) We have dropped the tensor product symbol for notational convenience although it is still implied. Note that we must be careful to preserve the ordering of the vectors and tensors and we must not confuse subscripts and superscripts. Otherwise, taking the gradient of a vector is straightforward. The result is a (1, 1) tensor with components µ A ν. But now µ µ! This is why we have introduced a new derivative symbol. We reserve the covariant derivative notation µ for the actual components of the gradient of a tensor. We note that the alternative notation A ν ;µ = µa ν is often used, replacing the comma of a partial derivative A ν,µ = µ A ν with a semicolon for the covariant derivative. The difference seems mysterious only when we ignore basis vectors and stick entirely to components. As equation (55) shows, vector notation makes it clear why there is a difference. Equation (55) by itself does not help us evaluate the gradient of a vector because we do not yet know what the gradients of the basis vectors are. However, they are straightforward to determine in a coordinate basis. First we note that, geometrically, µ e ν is a vector at x: it is the difference of two vectors at infinitesimally close points, divided by a coordinate interval. (The easiest way to tell that µ e ν is a vector is to note that it has one arrow!) So, like all vectors, it must be a linear combination of basis vectors at x. We can write the most general possible linear combination as µ e ν X Γ λ µν X e λx. (56) 20
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Welcome to Issue 2 of the Rad Report. Not all the campaigns are brand new, so if you've already seen them then feign interest for my ego's sake.
As the world appeared gripped in Olympic fever I was, of course, still watching MySkyed X-Games from a month or so ago.
Amongst the awesome skate, BMX and Big Air comps there was also one of the coolest sponsorships/ PR stunts I've seen in a while. I'm sure we all played with Hot Wheels at some point growing up (unsurprisingly I wasn't really a Barbie kind of girl), and the gravity-defying loop was always the centre of any decent track.
Hot Wheels added their own highly anticipated event at this year's X-Games by bringing on Rally car driver Tanner Foust and Hollywood stuntman Greg Tracy to set a world record by driving two Hot Wheels all-wheel-drive (isn't that the same as 4–wheel-drive?) rally cars through the 66-foot tall Hot Wheels Double Loop Dare track at X Games LA.
Branding heaven.
What 8 year old boy (and me) wouldn't want a Hot Wheels set after seeing this!
Check out the video: http://www.youtube.com/watch?v=c6PQ49B5Gpw
4 out of 5 revs for bringing the brand to life.
Here at Spark we work hard to make campaigns great, sometimes I'm sure you've all felt like you've put your blood sweat and tears in to make the campaign a success.
Serviceplan, a creative agency in Hamburg, took this phrase a little literally when recruiting new staff through a pretty out-there (and not entirely hygienic) billboard campaign.
Company copywriters donated blood-samples which were pumped around tubing on a billboard to create lettering. Designers sat in a mobile sauna and their sweat collected to be sprayed onto black fabric and the salt made the type. Thirdly 3kgs of raw onions were used to make staff cry and the teary tissues were fixed to the billboard.
A QR code on the billboards linked to a micro-site about Serviceplan for interested applicants (their form of 'TOGETHER' is a little creepier than ours).
Well this one is, um, 'creative'.
1 out of 5 for staff treatment, 5 out of 5 for potential contamination from blood-born pathogens.
Okay okay, sorry that one was a little wrong. I'll end on a positive note.
The Troy library in Michigan was running low on funds and close to closing; the local council proposed to vote in a 0.7% tax increase to keep the library open, but the issue became more about taxes than the library itself and caused great debate. With voting day near and negativity high about the tax increase the library officials took to social media in a controversial reverse-psychology WOM approach to change the public's mind.
The campaign was picked up locally, nationally and even internationally and was a great result for minimal budget.
The video is a couple of minutes long but worth a watch http://www.youtube.com/watch?v=UeOmgyWsksc
*Spoiler* the library didn't close. Told you there was a positive ending.
5 out of 5 hypothetical matchsticks for getting people talking.
Happy Friday everyone!


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Ormiston Gorge camping area
On the Larapinta Trail, this popular campsite can be reached by conventional 2WD vehicles and has good facilities. The site is 7 km off Namatjira Dr, 128 km west of Alice Springs. A number of walking trails start and finish here, including the Ormiston Pound Circuit and the Ghost Gum Lookout walk, which finishes at a viewing platform over Ormiston Gorge.
This campsite is located in West MacDonnell National Park
Stretching 170 km across the West MacDonnell Ranges west of Alice Springs, this is one of the most visited parks in the Red Centre. Its highest peak is 1531 m Mount Zeil, in the far west, and its gorges contain beautiful waterholes, startling white ghost gums, and retreats for wildlife. The park is connected by the 223 km Larapinta Trail, as well as by a multitude of shorter walks. A world-class bicycle path also leads through the park from Alice Springs. Most of the park trailheads are accessible by 2WD.
Want to see your campsite photos featured here? Send us your photos.
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Have you ever cruised the competition's website just to ogle their client roster? Sure you have. Not that you're interested in poaching; it's just natural curiosity. Are they working with big brands, or the next happening product?
I crave those juicy, high-profile projects as much as the next guy. But I can't let them define me any more than the bite-sized nuggets of work that land on my desk. In fact, there are a slew of advantages to being open to all kinds of opportunities, not just the ego feeders.
One-offs and quick-turns can move through your office rapidly and show clients just how high your level of service can be. You can react quickly to problems and have more time for research, brainstorming and planning -- adding to your cache and detracting from any repetitiveness that could weigh you down. You can meet with clients more often and work through several revisions. As a result, it's pretty easy to demonstrate your worth. And there's a rush of satisfaction that comes with getting good work done fast.
When I check my ego, I can bring in a greater variety of projects -- a great way to stave off burnout.
Boredom -- the soul-sucking kind that begs the question: "What's the point?" -- is to the brain what repetitive stress syndrome is to your joints and ligaments. Research supports the importance of variety -- the ultimate antidote to boredom -- in every aspect of life: diet, exercise, socializing, learning and work.
Taking on a variety of projects regardless of what your ego is screaming in the background -- big or small paycheck, national brand or not -- means more variety and less boredom, like a vitamin-B shot to your agency's collective brain trust.
Just don't mistake mundane for boring. Being bored because the task at hand feels like a wheel-spinning suck hole is not the same as performing necessary repetitive tasks, which have considerable value in the right doses. Not every hour of the work week can equate to go-go-go creative brilliance, or you'd burn out faster than a firecracker. More passive, "boring" activities such as reading or attending meetings have been shown to provide opportunity for daydreaming that results in increased creativity.
So don't be above the work. Design a web banner or a full-blown campaign with the same level of energy. If you're a brand manager, you realistically have a few projects a year that are big-budget, brag-sheet material. The rest of the time it's the lower-profile but equally important stuff that is just as important to your success as the client's. It's not just filler. If your ego is in check, it's keeping you sharp and engaged, and keeping your clients happy. They can tell when you're just running them through the mill.
Speaking of the mill mentality, we've all made the mistake of blabbing too much in the name of not being boring. We want to prove our worth and be validated as cool collaborators. But Catherine Blythe's "The Art of Conversation" tells us that listening is more important than talking. Make your conversations about the client, let them talk their faces off, glean lessons and provide feedback, and the remarkable thing is, you will be seen as interesting.
A final word on the boringness of ego: The thing that can stir any of us creatives into a tizzy is seeing our work watered down or completely reinterpreted by the client. Just remember that the work itself is still creative, even if the end product might not match your vision. You'll have some home runs that make your chest swell and some compromises that may shrink it. And maybe you need these lessons. Think about it: Whose vision matters anyway? Yours or the client's? You can try to steer them as wisely as you know how, but ultimately it's their product and their market. You're helping them solve their problems to make them look good, not you.
Until you score a benefactor with bottomless pockets and no opinions, you might as well embrace the goodness that is serving others. Because long-term success comes down to limited ego, with the not-too-shabby bonuses of more variety, more fun, and being able to fit your un-swollen head through the average doorway. It will keep you diversely skilled, uniquely attuned and maybe even a step ahead.


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Tagged: Austin Kearns
Cleveland Indians 2012 Offseason Recap and Preview
2011 was truly a fun year for the Cleveland Indians…the first half anyway:
Grade: B-
Notable Additions:
Casey Kotchman, Jeremy Accardo, Jose Lopez, Derek Lowe, Felix Pie, Kevin Slowey, and Dan Wheeler.
Notable Subtractions:
Jim Thome, Adam Everett, and Austin Kearns.
Why?: This is a weird entry/grade. The addition far surpass the subtractions. That said, I usually don't put in more than one or two players in the "Notable" lists that have been signed to a minor league contract. In this entry, however, I have five players that were signed to minor league contracts on the list. This serves as an indicator that a lot of the players aren't game changers. In other words, the subtraction players aren't that far away from equaling the value to a team that the addition players present, a fact which reflects itself in the grade I gave their offseason.
Also, Kosuke Fukudome and Chad Durbin have yet to sign, which may bring the Indians' grade down as they were on the Indians last year.
Predicted Record Range: 77-82 wins
New York Yankees Offseason Recap and Preview
Another year, another playoff. The only time the Yankees have missed the playoffs in my life time was in 2008 and they were closing a historic stadium:
As far as the other players lost go. Most were the product of mid season trades anyway. Meaning, the Yankees were not afraid to lose them. The Yankees will almost always have mid season acquisitions because they will almost always be in the hunt (how they always have trade pieces is another issue).
Predicted Record Range: 87-92 wins They will slow up a little. Maybe I'm just a spoiled Yankee fan but I think they will indeed contend once more. Another thing to take into account is the impact Jesus Montero will have on the line-up which I cannot account for.
To parents: recaps of our last two games will be up tomorrow. To everyone else: the study I mentioned in some entry I can't remember about adults' perception of baseball will be up before next Sunday.
Cleveland Indians Offseason Recap and Preview
Yeah, remember how in the recap of the Tiger's offseason, I said it's bad if you're remembered for a failure rather than your successes. Well, does anyone remember the Indians for anything else than failing at failure:
I mean really, can anyone think of anything else 'cause I've got nothing. I would say that this is a bit worse, than being know as the Tigers are that is. At least the Tigers got a great deal of good publicity but the Indians: Lost the game, almost got a perfect game thrown against them, messed up a perfect game by running a ball out, forever a side note, and still best know in 2010 for this error.
Grade: C
Notable Additions:
Austin Kearns, and Adam Everett.
Notablr Subtractions:
I tried looking up all the names of the player they had lost and saw nothing special about any in the group. Can't lose what you never had. Except for obviously, Carlos Santana to injury.
Why?: They didn't lose anything but didn't gain anything. Had they made their improvements and subtractions with bigger names I probably would have given them a better grade just because bigger names fire up the fan base more, but the Indians don't have the capital to accomplish this. If any doubt that the Indians lost no notable players, here are the names Drew Sutton, RJ Swindle, Winston Abreu, Luis Rodriguez, Chris Gimenez, Andy Marte, Wyatt Toregas. If you see any please tell me.
I like that they added two good players in Kearns and Everett, which is why I didn't give them a C-. Overall, not much to report on this front. They do look to develop with the forever anticipated and maybe only fantasized day that Travis Hafner returns to his former greatness as the second best DH in the game (this was when David Ortiz was hitting 40 HRs) and Carlos Santana develops into an above average catcher (he may be now but we haven't seen it because of the previously mentioned injury).
Predicted Record Range: 73-78 wins. This is again the mean for the possible scenarios that develop with Santana, Hafner etc. This means the range could go to 75-80 if everything goes right or they could actually lose more games because of their lack of improvements and the much improved AL Central.
Next up: Kansas City Royals
Again, well see when it comes to today's double header at Fordham. One thing to look forward to (for some people, I get that not all of you like Spring Training much less High School Baseball) is that these following entries *WILL* have pictures! So no more pictures showing how awesome are field is:
but look at it. This is college level field where I have my own luxury suite (more on that in the entry). Where was I going with this point again? Oh… yeah, the following entries will be much more extensive and more visually appealing than there outdated counterparts.


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By John Fallon
Seven months before Ireland's Euro 2016 qualifier against Scotland, the shadow boxing has already started at Stoke City.
Marc Wilson, one of three Ireland internationals in the Potters' ranks, has revealed Stoke's sole Scot Charlie Adam is on a one-man crusade to unsettle the Irish.
But Antrim man Wilson remains adamant the qualifier's likely venue of Parkhead is just one reason why Ireland hold the
"Charlie's been dishing it out to us Irish players since the draw," says Wilson with a grin.
"He reckons their Scotland side are better than us but I can't agree.
"It's going to be a great game, especially if it's at Parkhead, and November won't be long coming around now."
In the meantime, the pressing international duties come in May and June with four friendlies.
His brother's wedding rules Wilson out of the second game on May 31 against Italy but he's told Martin O'Neill and Roy Keane to count him in for the rest.
"I'm best man at my brother's wedding, so he'd have my life if I wasn't there," he said.
"I should be available for the rest of the matches and they'll be important preparation."
Wilson (27) has revelled under the new managerial ticket, nestling into position as one of the nailed-on centre-backs for the campaign ahead.
O'Neill was effusive in his praise for the defender following the scoreless draw in Poland in November, albeit cautionary about Wilson's excessive confidence on the ball.
Playing out from the back is a trait encouraged by Stoke manager Mark Hughes but Wilson has no hesitation following his international manager's orders.
"He (O'Neill) did have a word with me about that," said the former Portsmouth captain.
"I probably know myself that it's a different style from Stoke. If the ball has to go in row Z with Ireland, it will go in row Z.
"We've been given a new lease of life in the Ireland squad under Martin and Roy.
"I think we got tired in the last game against Serbia because of all the club matches in March but the squad is shaping up nicely."
Patience has been required on the part of Wilson to realise his dream.
After jumping ship from Northern Ireland at the age of 15, he rarely got a look-in at Under-19 and Under-21 level. Giovanni Trapattoni was also slow to appreciate the talents of a player until the 6-1 hammering by Germany in October 2012 opened the door.
"I was only picked once in the Ireland's Under-21s squad," he said. "That was in Luxembourg in 2007 when I played as a second-half sub and set up a goal for Anthony Stokes.
"I just wasn't called up after that and was never given a reason why. Yes, it was a big gap after that until the senior call came but there was never any chance of me going back to play for Northern Ireland. Representing the Republic was always my ambition."