Espacio, Tiempo y materia en el Cosmos J. Alberto Lobo ICE/CISC-IEEC Summary Part I: General Relativity Theory • • • • • Concepts of motion: From Aristotle to Newton Newton’s laws: Absolute Space and Time Mach’s criticism and Mach’s Principle Einstein’s Equivalence Principle General Theory of Relativity: – Curvature of space-time – Field equations: Space, Time and Matter link together Part II: General Relativistic Cosmology • • • • • • • Newtonian Cosmology, Olbers’ paradox General Relativistic Cosmology, an evolutionary paradigm The Cosmological Principle Friedmann equations Big Bang Cosmology: parameters of the Universe The Accelerating Universe Outlook and future talks A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 2 Concepts of motion Aristotle expressed in his treatise “Physica” that a force is necessary to keep bodies in motion, no matter whether accelerated or not. Galileo Galilei, some 1800 years on, proved by experiment that uniform motion does not require a force. And that forces cause accelerations. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 3 Newton’s laws of Mechanics Isaac Newton wrote a fundamental book in 1689: “Principia Mathematica Philosophiae Naturalis”. There, he developed his theories of motion and of Universal Gravitation. Newton’s laws of mecahnics are explained in High School courses. We recall them here now: 1. 2. Under no forces, bodies move at constant speed, or stay at rest. Forces cause accelerations, according to the equation: F = ma 3. Every force action has a reaction counterpart, equal and opposite. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 4 Let’s review the 1st law Is it a consequence of the second? For, it may be argued, if F = ma , and F = 0 ⇒ a = 0 and v = constant Problem is: how can one possibly asceratain that F=0? This is in fact a logically circular question: F = 0 R v = constant The first law actually defines the class of reference systems in which the second law applies. These are called Inertial reference systems. Newton inferred from here the existence of Absolute Space (Scholium of his PM), and that it was the cause of inertia. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 5 The bucket of water experiment Rope wound Rope unwinds Water at rest A Lobo, 14-ii-2007 Water rotates Espacio, Tiempo y Materia en el Cosmos 6 Inertia and Absolute Space Newton derived important conclusions from his experiment: Accelerated motion is an objective concept. 1. There is an absolute reference system which enables the intrinsic definition of acceleration. 3. This is Absolute Space. It is therefore the causal agent of inertia. 4. It cannot however be individuated. 5. “Fixed stars” are (perhaps) the best approximation... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 7 For the next 200 years... D. Papin J. Watt U. Leverrier P.S. Laplace I. Kant L. Boltzmann Immense success of Newtonian Mechanics and its variety of applications somewhat left in stand by those almost philosophical discussions: • • • • Mechanical machines of many types: industry, etc. Celestial Mechanics Mechanical theory of Thermodynamics ... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 8 Ernst Mach’s criticism What would happen if the rest of the Universe should start revolving around the bucket? He conjectured the same would happen... Mach went on to formulate what was to become known as Mach’s Principle: –The inertia of the bodies is determined by the distribution of matter throughout the Universe. Unfortunately, Mach’s principle is not any quantitative... Ernst Mach A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 9 General Relativity The old riddle was solved by Einstein’s General Theory of Relativity, towards the end of 1915. 1 8π G Rµν − R gµν = − 4 Tµν 2 c 1950 Because that solution heavily relies on Gravitation –it actually constitutes the new theory of gravitation–, it will be useful to review a few basic facts about gravity... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 10 Bodies in free fall A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 11 Port Aventura A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 12 Simple theoretical analysis Rope tension: T Rope tension: T F = Ma ≠ 0 ⇓ a=g≠0 T +F = 0 ⇓ a=0 Gravity force: F A Lobo, 14-ii-2007 Gravity force: F Espacio, Tiempo y Materia en el Cosmos 13 A very contemporary example of Galileo’s experiment: The astronaut is in free fall at fhe same rate as the spacecraft, so he feels weightlessly floating. Nevertheless, gravity up there is not negligible: 8 m/sec2... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 14 “Hipotesis non fingo” Let h = 5 m v h L L = vt 1 2 h = gt 2 ⇒ L = 2hv 2 / g ⇒ A Lobo, 14-ii-2007 v = 700 m/s L = 700 m Espacio, Tiempo y Materia en el Cosmos v = 100 km/s L = 100 km 15 “Hipotesis non fingo” But take a careful look at these numbers: v = 100 km/s L = 100 km ^^^^^^^^^ 100 km >100 km !!! The ocean’s surface actually bends… If the Earth were flat then L=100 km But the Earth is not flat!! Hence L > 100 km !!! A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 16 “Hipotesis non fingo” Is it thinkable to shoot the bullet fast enough that it eventually hits the pirate from the back ??? Like this: mmm... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 17 “Hipotesis non fingo” Perhaps the Moon is, after all, an example of that...: Somebody must have shot it from where it revolves around the Earth... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 18 “Hipotesis non fingo” Newton’s Law of Universal Gravitation v=0 v small v large GMm F= 2 r A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 19 Let’s pause for a moment... ...and write down the two main results seen so far: 1. The definition of Inertial Reference System is circular, therefore Newton’s laws –and Special Relativity laws, too— suffer from a severe logical inconsistency. 2. Gravitational fields impinge the same acceleration on all bodies, regardless of their mass and/or composition. Inertial Reference Systems may not have a prvileged status, any Reference System should be equally appropriate to describe Physical laws. General, rather than Special Relativity must be the ultimate concept in Physics. It is not possible to tell whether a given Reference System is accelerated by a gravitational field or by some other agent. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 20 Equivalence Principle In 1915, Einstein came up with a far reaching conjecture which settled the basis for a new theory of gravity, and disposed of the epistemological inconsistencies of Newtonian Mechanics. The Equivalence Principle The laws of Physics are those of Special Relativity when expressed in a Reference System which is falling freely in a gravitational field. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 21 Equivalence Principle -- The EP disposes of the servitudes of needing Inertial Systems. General Relativity is soundly established instead. -- Inertial and gravitational masses must be equal. -- Absolute space is a superfluous concept. -- Freely falling particles (generally) follow curved trajectories. These are straight lines if no gravitational fields are present, and shortest distance curves (or geodesics) if they are. -- Space-time is therefore not absolute, it has geometric properties determined by the distribution of gravitating mass and energy. -- The old tenet of Euclidean Geometry thus gives way to the rules of Riemannian Geometry. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 22 Field equations and geodesics The EP sets the general framework. But a very important question is still pending: How does matter distribution actually determine geometry? The answer is provided by Einstein’s field equations: 1 8π G Rµν − R g µν = − 4 Tµν 2 c which is a complicated set of non-linear partial differential equations for the field unknowns gµν. Once the gµν’s are known, the trajectories of test particles and light rays are determined by the geodesic equation: x µ + Γ µρσ x ρ xσ = 0 A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 23 Bending of light rays A truly remarkable prediction of GR is the bending of light rays: just like material particles, light rays (photons) follow non-straight line trajectories as they come close to gravitating objects. Take for example a ray which grazes the Sun surface: 4GM : Einstein’s formula: δφ = 1.76 arc - seconds 2 Rc A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 24 Eddington’s 1919 observations Night A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 25 Eddington’s 1919 observations Day A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 26 Solar Eclipse A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 27 Gravitational lenses A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 28 End of Part I A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 29 Summary Part I: General Relativity Theory • • • • • Concepts of motion: From Aristotle to Newton Newton’s laws: Absolute Space and Time Mach’s criticism and Mach’s Principle Einstein’s Equivalence Principle General Theory of Relativity: – Curvature of space-time – Field equations: Space, Time and Matter linked together Part II: General Relativistic Cosmology • • • • • • • Newtonian Cosmology, Olbers’ paradox General Relativistic Cosmology, an evolutionary paradigm The Cosmological Principle Friedmann equations Big Bang Cosmology: parameters of the Universe The Accelerating Universe Outlook and future talks A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 30 Cosmology: Preamble Cosmology is concerned with the entire Universe –a big problem indeed, and a very old one, too... The scientific approach to Cosmology is however much younger, about 300 years only. Newtonian Cosmology faces a number of insurmountable problems: • Space must be infinite: what would be there beyond its limits otherwise? • Likewise, time must be infinite: the Universe is thus eternal. • As a consequence, the gravitational potential is also infinite. This creates a problem of stability versus infinitely long life. • Olbers’s paradox: the sky is dark at night... General Relativistic Cosmology fixes these issues because it attributes dynamic properties to space and time. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 31 Contemporary Cosmology Modern Cosmology –so called Standard, or Big Bang model– is based upon a few major observational facts: • • • • The Universe is very highly isotropic –at large scales. The Universe is expanding at the present epoch. The Universe is filled with thermal radiation of ~2.7o Kelvin. Abundance of light elements: H—73%, He—26%. The Standard Model rests on two fundamental hypotheses: • General Relativity theory • Cosmological Principle. It also needs many other sources of scientific knowlwdge, such as High Energy Physics, Chemistry, Thermodynamics, Astrophysics, Statistics, etc. Cosmology is definitely a very pluri-disciplinar activity. This talk addresses only the “GR connection”. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 32 The Cosmological Principle The Cosmological Principle is the assumption that: 1. 2. The Universe is large scale isotropic. Us, humans, are standard observers of the Universe as regards its large scale properties. Hence the Universe is: A Lobo, 14-ii-2007 ⎧isotropic ⎪ ⎨ and ⎪⎩homogeneous Espacio, Tiempo y Materia en el Cosmos 33 Friedmann-Robertson-Walker The CP has an immediate mathematical consequence as to which is the large scale geometry of the Universe: ⎡ dr 2 2 2 2 2 ⎤ + r (dθ + sin θ dϕ ) ⎥ ds = c dt − a (t ) ⎢ 2 ⎣1 − kr ⎦ 2 2 2 2 (FRW) where a(t) is the scale factor of the Universe, and k is a trichotomic constant: a (t ) > 0 Universe expand a (t ) < 0 Universe A Lobo, 14-ii-2007 s shrinks ⎧+1 closed, spherical ⎪ k = ⎨ 0 open, flat ⎪⎩−1 open, hyperbolic Espacio, Tiempo y Materia en el Cosmos 34 The expanding Universe Whether the Universe actually expands or shrinks, i.e., whether a (t ) > 0 or a (t ) < 0 is to be determined by observation. It appears that, at the current epoch, the Universe is in an expansive phase. Time advances in this direction FRW geometry A Lobo, 14-ii-2007 This is determined by the fact that light from remote galaxies arrives red-shifted to us. Espacio, Tiempo y Materia en el Cosmos 35 Cosmological red-shift Cosmological red-shift Here Far away Even farther λreception − λemission z= , λemission A Lobo, 14-ii-2007 measure of redshift Espacio, Tiempo y Materia en el Cosmos 36 Cosmological red-shift This was fisrst discovered in 1929 by Edwin Hubble, who interpreted redshift as an indication that galaxies move away from us. Hubble also inferred that the expansion speed is proportional to distance: v = H 0d , Hubble’s law where H0 is known as Hubble’s constant: H 0 75 km / sec⋅ Mpc In the light of FRW geometry, redshift is an evolutionary effect: z= a (treception ) a (temission ) −1 Redshift is therefore a measure of age and size of the evolving Universe A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 37 The geometry of space k = +1 k=0 k = −1 A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 38 Friedmann equations The evolution of the Universe is described by Einstein’s equations: 1 8π G Rµν − R g µν + Λg µν = − 4 Tµν c 2 where Λ is the so called Cosmological Constant. Tµν includes both ponderable matter and radiation in the Universe. Two magnitudes characterise their contribution: density and pressure. Expanding Einstein’s equations one finds Friedmann’s equations. In the present time, the Universe is matter-dominated, i.e., pressure is negligible. The following obtains: a 2 8πGρ k Λ , H ≡ 2 = + 2+ a 3 a 3 2 A Lobo, 14-ii-2007 p ρc 2 Espacio, Tiempo y Materia en el Cosmos Friedmann equation 39 Λ=0 Cosmological evolution ρ0 < ρc ρ0 = ρc ρ0 > ρc A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 40 Λ=0 Cosmological evolution There is a critical density, which determines whether the Universe is spherical, hyperbolic or flat: ⎧ k = +1 if ρ 0 > ρ c spherical 3H 02 ⎪ ρc = , ⎨ k = 0 if ρ 0 = ρ c flat 8π G ⎪ k = −1 if ρ < ρ hyperbolic 0 c ⎩ But we do not know which of the three is true, as measuring ρ0 is a very difficult task. Indeed, under forms of matter... ρ0 go both visible and dark To complicate things further, the Universe was recently seen to be under accelerated expansion!! A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 41 Parameters of the Universe It is customary to define dimensionless parameters which characterise the evolution of the Universe in terms of the critical density. They are: ΩM = 8πGρ0 3H 02 matter ΩΛ = Λ 3H 02 dark energy, or Λ Ωk = k H 02 Curvature The Friedmann equation is then rewritten as: 3 2 H 2 ( z ) = ⎡Ω M (1 + z ) + Ω k (1 + z ) + Ω Λ ⎤ H 02 ⎣ ⎦ and gives rise to complicated casuistics, as we see next... A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 42 Possible states of the Universe A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 43 Composition of the Universe We live in a Universe which is: • • • • • • • Spatially flat Therefore of infinite volume In accelerated expansion Dominated by dark energy With only 4% of visible matter 13 billion years old Will expand forever This model is certainly not free of problems... To discuss them is the objective of this series of talks. A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 44 A Lobo, 14-ii-2007 LISA and the GW Universe Espacio, Tiempo y Materia en el Cosmos 45 End of Presentation A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos 46