The trampoline of gravity

courtesy of Analogies are an important part of understanding, as well as the popularisation of physics.

However, analogies are analogies and at some point always fail to capture the full picture of what is going on. More than that, taking analogies too seriously can lead to misunderstandings.

Einstein’s general relativity basically tells us that massive objects bend the space-time they are sat in and that this is the origin of gravity. To really understand this one has to pull apart the Einstein field equations in all their tensorial beauty. I won’t do that here and now.

A common analogy here is that of a heavy bowling ball placed on a trampoline. The bowling ball deforms the elastic trampoline surface, it sags, and this is similar to how a massive object, say a star, bends the space-time around it.

One can now “model” photons or test particles by using light balls, say ping-pong balls. The point is that these near weightless balls will not deform the trampoline’s elastic surface. When the bowling ball is not on the trampoline the light balls move in straight lines when given a light initial push. When the bowling ball is on the trampoline the light balls no longer follow straight lines, but curved paths. These light balls are attracted to the bowling ball: thus we have gravity!

This is a great analogy for light rays or photons in general relativity. Light is bent around massive objects like stars. If you have access to a trampoline and some heavy and light balls, play around and experiment for yourself.

However, This analogy seems to be the principle source of misunderstandings and even scepticism of general relativity for the untrained.

Conceptionally the analogy breaks down because the trampoline does not represent the three dimensional space we inhabit, or rather a time slice of our four dimensional world. All we have is an embedding of a two dimensional geometry in our three dimensional flat world.

The trouble is that the space-time of general relativity does not require any such embedding in a higher dimensional flat space. It is of course true that mathematically we can always find (isometric) embeddings in higher dimensional spaces of the geometries found in general relativity, but this does not imply that nature uses such things.

The other issue is that the trampoline analogy really misses the curvature of space-time and only highlights space curvature. The ping-pong balls move about the sheet being “attracted” to the bowling ball because of the depression in the elastic sheet. The trouble is that in general relativity test particles, say photons, move in the “straightest possible path” in four dimensions, including time. This fact is missed by the analogy.

So however useful this analogy is, and I say it is useful, it cannot really describe general relativity. Objections, philosophically or otherwise to general relativity cannot be founded on the trampoline analogy.

The great man himself. Moreover, there is plenty of direct and indirect experimental verification that general relativity is a good model of gravity. This fact seems rather inescapable: there are no consistent repeatable experiments that, taking into account the domain of applicability and experimental errors, that suggest that general relativity is not a good model. I may say more about this another time.

In short, love analogies, use analogies, tell other people about analogies, however remember they are analogies and no replacement for mathematical models.

Quantum Field Theory A Modern Introduction by M. Kaku

Quantum Field Theory: A Modern Introduction

Quantum field theory is a many faceted subject and represent our deepest understanding of the nature of forces and matter. Quantum field Theory A Modern Introduction by Michio Kaku gives a rather wide overview of many essential ideas in modern quantum field theory.

The readership is graduate students in theoretical physics who already have some exposure to quantum mechanics and special relativity.

The book is divided into three parts.

Part 1 Quantum Fields and Renormalization

Chapter 1 gives a historic overview of quantum field theory. Topics here include: a review of the strong, weak and gravitational interaction, the idea of gauge symmetry, the action principle and Noether’s theorem.

Symmetries and group theory are the subjects of Chapter 2. Topics include: representations of U(1), SO(2), SO(3) and SU(2), spinors, the Lorentz group, the Poincare group and supersymmetry.

Chapter 3 moves on to the quantum theory of spin-0 and spin 1/2 fields. The emphasis here is on canonical quantisation. Topics covered here include: the Klein-Gordon field, propagator theory, Dirac spinors and Weyl neutrinos.

Quantum electrodynamics is the topic of Chapter 4. Again the emphasis is on canonical quantisation. Topics include: Maxwell’s equations, canonical quantisation in the Coulomb gauge, Gupta-Bleuler quantisation and the CPT theorem.

Chapter 5 describes the machinery of Feynman diagrams and the LSZ reduction formula. Topics here include: cross sections, propagator theory, the LSZ reduction formulas, teh time evolution operator, Wick’s theorem and Feynman rules.

The final chapter of part 1, Chapter 6 describes the renormalization of quantum electrodynamics. Topics here include: nonrenormalizable & renormalizable theories, the renormalization of phi-4 theory, regularisation, the Ward-Takahashi identites and overlapping divergences. The renormalization of QED is then broken down into fours steps.

Part 2 Gauge Theory and the Standard Model

Chapter 8 introduces path integrals which are now fundamental in particle theory. Topics here include: path integrals in quantum mechanics, from first to second quantisation, generators of connected graphs, the loop expansion, integration over Grassmann variables and the Schwinger-Dyson equations.

Chapter 9 covers gauge theory. Topics here include: local symmetry, Faddeev-Popov gauge fixing, the Coulomb gauge and the Gribov ambiguity.

The Weinberg-Salam model is the subject of Chapter 10. Topics here include: broken symmetries, the Higgs mechanism, weak interactions and the Coleman-Weinberg mechanism.

Chapter 11 discusses the standard model of particle physics. Topics here include: the quark model, QCD, jets, current algebra, mixing angles & decays and the Kobayashi-Maskawa matrix.

Chapter 12 discusses anomalies and the Ward identities. Topics here include: the Ward-Takahashi identity, the Slavonov-Taylor identities, BRST symmetry & quantisation, anomalies and Fujikawa’s method.

Chapter 12 covers the remormalization of gauge theories. Topics include: counterterms, dimensional regularization and BPHZ renormalization.

The modern perspective of QFT is based on Wilson’s renormalization group. Chapter 14 introduces the reader to this concept in the context of QCD. Topics here include: deep inelastic scattering, neutrino sum rules, the renormalisation group, asympptotic freedom and the Callan-Symanzik relation. The renormalization of QCD is presented via renormalization groups methods.

Part 3 Nonperturbative Methods and Unification

Chapter 15 introduces lattice gauge theory which allows questions in quantum field theory to be numerically tackled on computers. Topics here include: the Wilson lattice, scalars & fermions on the lattice, the strong coupling approximations, Monte Carlo simulations and the renormalization group.

Topological objects in field theory are the topic of Chapter 16. Topics include: solitons, monopoles, instantons & tunneling and Yang-Mills instantons & the theta vacua.

Chapter 17 discusses phase transitions and critical phenomena. Topics covered include: critical exponents, the Ising model, the Yang-Baxter relations, the mean-field approximation and scaling & the renormalisation group.

The idea of unification is the subject of Chapter 18. Topics include: unification & running coupling constants, SU(5), anomaly cancellation, the hierarchy problem, SO(10), technicolor, preons & subquarks and supersymmetry and strings.

Chapter 19 discusses quantum gravity. This chapter is about attempting to construct a perturbative theory of quantum general relativity. Topics include: the equivalence principle, vierbeins & spinors, GUTs & cosmology, the cosmological constant, Kaluza-Klein theory and counter terms in quantum gravity.

Supersummetry is the subject of Chapter 20. Topics covered here include: supersymmetric actions, superspace methods, Feynman rules, nonrenormalization theorems, finite field theories, super groups and supergravity.

Chapter 21 introduces the superstring. Topics include: quantisation of the bosonic string, teh four superstring theories, higher loops, string phenomenology, light-cone string field theory and the BRST action.

The book contains exercises.

Paperback: 804 pages
Publisher: OUP USA; New Ed edition (6 Oct 1994)
Language English
ISBN-10: 0195091582
ISBN-13: 978-0195091588