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Relativity and Cosmology I Introduction to Relativity and Cosmology I J. Mark Heinzle Gravitational Physics, Faculty of Physics University of Vienna Version 20/01/2010 –2– version 20/01/2010 CHAPTER 1 AETHER 1.1 Galilean space (...and time) What is space? Which mathematical concept provides the best description for what we experience as our spatial world? This question is neither naïve nor trivial. There exists a multitude of concepts in mathematics that could apply a priori. Space could be merely a set, it could be a topological space, maybe Hausdorff, a semi-group, a field, a manifold, a complex manifold, a vector space, . Our daily experience tells us that space is some kind of ‘continuum’ of ‘points’. It is a matter of course for us to be able to form ‘arrows’ (‘vectors’) −→pq connect- ing points p,q in space. Most importantly, we can add vectors and multiply vectors with scalars ( R). This suggests that our spatial world has a struc- ∈ ture intimately connected with the structure of a vector space. However, it is obvious that a vector space is not a good model for our spatial world. This is simply because there doesn’t seem to exist a distinguished zero vector; quite the contrary, all spatial points seem to be on an equal footing. Collecting these intuitive observations results in the following statement: Space is an affine space over a three-dimensional real vector space. version 20/01/2010 –3– Galilean space (...and time) Chapter 1. Aether The affine space we live in is endowed with a geometric structure that is so fundamental that it is easily overlooked. We can measure the length of vectors (e.g., in meters) as well as angles between vectors. The mere existence of the concepts of length and angles entails that the vector space underlying our affine space is a Euclidean vector space, i.e., it carries a scalar product. Space is an affine space over a three-dimensional Euclidean vector space. Definition. We call this model of our spatial reality Galilean space. Remark. All these “experiments” that we perform every day—we form vectors connecting points, compute lengths and angles— indicate that Galilean space is the correct mathematical description of the space of our daily experience. Note, however, that we would come to a radically different conclusion if the scale of our intuitive perception were different. But we think in meters and not in megaparsecs. What is time? This is a somewhat more difficult question. Let us quote Sir Isaac Newton: “Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time . ” Summarizing, we don’t quite know what time is (at least not yet), but we have a lot of first hand experience with it. For our present purposes this is sufficient. In order to formulate the laws of physics it is essential to choose a coordi- nate system.1 A coordinate system enables us to make measurements; we can quantify position by real numbers, and we are able to perform a mathematical analysis of our physical fields: we form derivatives, gradients, divergences and formulate (differential) equations for our physical unknowns. We are free to choose any curvilinear coordinate system we can come up with.2 1Strictly speaking, a coordinate system—or chart—is simply a bijective map of R3 into Galilean space. In the present context, however, dependence on time is permitted (so that a coordinate system is a one-parameter family of maps of R3 into Galilean space). 2And frequently, curvilinear coordinates are very useful. For instance, when one aims at studying rotating bodies, spherical coordinates are often advantageous. –4– version 20/01/2010 Chapter 1. Aether Galilean space (...and time) However, among the vast number3 of possible coordinate systems, there exists a special subfamily: the inertial frames (inertial coordinates). The existence of these inertial frames is guaranteed by the principle of relativity: Special principle of relativity There exists a family of coordinate systems, which we call inertial frames of reference, w.r.t. which the laws of nature take one and the same form. In other words, the mathematical formulation of the laws of physics is identi- cal for all ‘inertial observers’. In the Galilean context, an inertial frame is thus a coordinate system of Galilean space, for which the laws of (Newtonian) physics assume their conventional form. Consider, for example, Newton’s first law which describes the motion of a point particle in the absence of exterior forces. Newton’s first law states that these point particles move along straight lines in Galilean space. In an inertial frame of reference, Newton’s first law is represented by the equation4 x¨i(t) = 0 . Another law of (Newtonian) physics is the Poisson equation describing the gravitational field of a massive body. In an inertial frame of reference we have ∆φ = ρ , where φ is the potential and ρ the matter density. The Laplacian ∆ is given by ∂ 2 ∂ 2 ∂ 2 ∆= + + = ∂2 + ∂2 + ∂2 = δij ∂ ∂ . ∂x1 ∂x2 ∂x3 1 2 3 i j In a general curvilinear coordinate system, e.g., in spherical coordinates, a straight line does not appear as straight; consequently, the differential equation representing Newton’s first law is different. (Of course, the same is true for the Poisson equation.) The price we pay by not choosing an inertial coordinate system is that the laws of physics take a more complicated form (that includes terms that are interpreted ficticious forces). 3The vast number is of course ∞. 4A point particle is represented by a curve t 7→ x(t); the components of the vector x are denoted by xi, i = 1, 2, 3. This curve is a straight line if and only if x¨(t) = 0 (or, in component notation, x¨i(t) = 0). version 20/01/2010 –5– Galilean transformations Chapter 1. Aether 1.2 Galilean transformations Definition 1.1. A Galilean transformation is a coordinate transformation5 of Galilean space that maps one inertial frame into another. Taking account of the definition of an inertial frame we find an alternative definition. 6 Definition 1.1′. A Galilean transformation is a coordinate transformation of Galilean space that leaves invariant the laws of (Newtonian) physics. Remark. We emphasize that the principle of relativity implies that all laws of (Newtonian) physics are invariant under Galilean transformations. On the basis of definition 1.1′ we are able to derive the Galilean transformations; we choose to make use of Newton’s first law. Suppose that (t,x), where x = (x1,x2,x3), and (t,¯ x¯), where x¯ = (¯x1, x¯2, x¯3), are inertial coordinates. By definition, the Galilean transformation mapping the one inertial coordinate system into the other is of the form7 t¯ = t , x¯ = ¯x(t,x) ; (1.1) expressed in component form, the latter reads x¯i = ¯xi(t,x1,x2,x3). In (1.1) we assume that the origin of space and time remain unchanged; if this is not the case, (1.1) may contain additional constants representing a spatial translation and a temporal translation, i.e., t¯ = t + t¯0 , x¯ = ¯x(t,x) +x ¯0 . (1.1′) Let x(t) represent the freely moving point particle w.r.t. the first inertial frame, i.e., x¨i(t) = 0. W.r.t. the second inertial frame (1.1), this curve reads 5Strictly speaking, a Galilean transformation is a one-parameter family of coordinate trans- formations of Galilean space; this is because there might be a dependence on time, see (1.1). 6See footnote 5. 7It is implicit in definition 1.1′ that time remains unchanged under a Galilean transformation (because it is a coordinate transformation ‘of Galilean space’). –6– version 20/01/2010 Chapter 1. Aether Galilean transformations x¯(t)= ¯x t,x(t) ; we may replace t by t¯. We obtain ∂¯xi ∂¯xi x¯˙ i = + x˙ j , ∂t ∂xj ∂2¯xi ∂2¯xi ∂2¯xi ∂¯xi x¯¨i = + 2 x˙ j + x˙ jx˙ k + x¨j . ∂t2 ∂t∂xj ∂xj∂xk ∂xj |{z} =0 Since Newton’s law must be invariant under the transformation (because (t,¯ x¯) is assumed to be an inertial frame) we find that ∂2¯xi ∂2¯xi ∂2¯xi 0= + 2 x˙ j + x˙ jx˙ k . ∂t2 ∂t∂xj ∂xj∂xk Since x˙ i is arbitrary, this results in ∂ ∂¯xi ∂ ∂¯xi ∂ ∂¯xi ∂ ∂¯xi = 0 , = 0 and = 0 , = 0 , ∂t ∂t ∂xj ∂t ∂t ∂xk ∂xj ∂xk from which we conclude that ∂¯xi ∂¯xi vi = const and Ri = const ∂t ≡− ∂xk ≡ k Accordingly, (1.1) becomes t¯ = t , x¯i = Ri xk vi t ; (1.2) k − in matrix notation the latter reads x¯ = Rx vt. − The matrix R is not arbitrary. It is straightforward to prove that R O(3), ∈ i.e., R is either a rotation (R SO(3)) or a reflection (R O(3), det R = 1). ∈ ∈ − (Note, however, that this does not follow directly from the invariance of New- ton’s first law—it is necessary to invoke another law of physics.) Exercise. Use the invariance of the Poisson equation under (1.2) to prove that R O(3). ∈ In (1.2) the origin of space and time remain unchanged; if this is not the case, (1.2) may contain additional constants representing a spatial translation and a temporal translation, i.e., t¯= t + t¯ , x¯i = Ri xk vi t +x ¯ . (1.2 ) 0 k − 0 ′ version 20/01/2010 –7– Galilean transformations Chapter 1.
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