Introducing General Relativity by Mark Hindmarsh Andrew Liddle Book Read Online And Epub File Download
Overview: Introducing General Relativity
An accessible and engaging introduction to general relativity for undergraduates
In Introducing General Relativity, the authors deliver a structured introduction to the core concepts and applications of General Relativity. The book leads readers from the basic ideas of relativity—including the Equivalence Principle and curved space-time—to more advanced topics, like Solar System tests and gravitational wave detection.
Each chapter contains practice problems designed to engage undergraduate students of mechanics, electrodynamics, and special relativity. A wide range of classical and modern topics are covered in detail, from exploring observational successes and astrophysical implications to explaining many popular principles, like space-time, redshift, black holes, gravitational waves and cosmology. Advanced topic sections introduce the reader to more detailed mathematical approaches and complex ideas, and prepare them for the exploration of more specialized and sophisticated texts.
Introducing General Relativity by Mark Hindmarsh Andrew Liddle Book Read Online Chapter One
Introducing General Relativity
It is now more than a hundred years since Albert Einsteinpresented the final form of the General Theory of Relativity to the Prussian Academy of Sciences, in November 1915. Since then, it has migrated from an extraordinary achievement at the frontiers of physics, reputedly understood by only a very few, to a standard advanced undergraduate course. General Relativity (as it is usually called, commonly shortened to simply GR) is essential for the understanding of the Universe as a whole, wherever gravity is strong, and also whenever precise time measurements are made. The Global Positioning System (GPS), now built in to billions of devices around the world, would not work without the General Relativistic prediction that clocks run more slowly on Earth than in the satellites defining the GPS reference frame.
Part of the fascination of General Relativity lies in the personality of Einstein, and the way he is often presented as a lone genius working for years in isolation, finally to reemerge with the fully formed and beautiful theory we know today. In reality, he was in constant communication with other scientists, and others were working towards a relativistic theory of gravitation. The first such theory was actually written down by Gunnar NordstrÃ¶m in 1913, who attempted a direct relativistic generalisation of the Newtonian gravitational potential. Einstein was the first to understand that the appropriate dynamical quantity is the space–time metric itself, but the geometric aspect of General Relativity was probably first appreciated by the mathematicians Marcel Grossmann and David Hilbert. Einstein worked with his friend Grossmann, and had crucial correspondence with Hilbert before coming up with the final and correct formulation. Einstein himself took several wrong turnings doing the years between 1907 and 1915 when he was working most intensively on the theory. The lone genius is a myth, but it is fair to consider that General Relativity is Einstein's own, and crowning, achievement.
The technical complexity of General Relativity comes from several sources. The fundamental objects of relativity are tensors, because the relativity principles (both special and general) are statements about the properties of physical laws under transformations between coordinates. Thus any General Relativity course must start with tensor calculus. General Relativity is a geometrical theory, treating space–time as a manifold, describing its dynamics in terms of geometrical quantities. Thus in approaching General Relativity the basic geometrical concepts developed by Bernhard Riemann and others from the mid 1800s — of connection, geodesic, parallel transport, and curvature — must be introduced. Tensor calculus and Riemannian geometry are not part of the standard mathematical equipment of a physics undergraduate. This was also true in Einstein's undergraduate career, although there were courses on offer. It has been speculated that had he gone to an advanced geometry course, he would have later saved himself several years’ work.
A final difficulty is the one of translating the mathematical concepts into physical observables. General Relativity rethinks the fundamentals of space and time, which take part in physical processes rather than being a framework on which things happen. So deciding what is observable, rather than simply an artefact of a particular choice of coordinates, is difficult. Indeed, Einstein changed his mind a couple of times as to whether gravitational waves were real or not, and it took about fifty years for a unanimous view to emerge.
Gravitational waves, an early prediction of General Relativity, are amongst the hottest topics in physics following their direct detection by the LIGO/Virgo collaboration, announced in 2016. Their real significance is not so much as a triumphant vindication of Einstein's theory; there was no serious doubt that gravitational waves existed following the careful measurements of the orbital decay of a binary pulsar system discovered by Russell Hulse and Joseph Taylor in the 1970s. Rather, the detection signals the beginning of a new branch of astronomy, which has the prospect of detecting violent astronomical events right back to the very earliest stages of the Big Bang. New detectors, similar to LIGO and Virgo, are being built in Japan and India, and a space‐based gravitational wave detector called LISA is planned for the early 2030s. General Relativity will continue to be at the forefront of scientific research in the 21st century, as it was throughout the 20th.
Chapter 2
A Special Relativity Reminder
The Special Theory unites space and time dot shorter lengths and longer times dot seeing it with diagrams
Before launching into our account of General Relativity, we give a brief reminder of the main characteristics of its predecessor theory, the Special Theory of Relativity. This was introduced by Einstein in 1905, and is usually referred to by the shorthand Special Relativity. These theories have a rather different status to traditional physics topics, such as electromagnetism or atomic physics, which seek to understand phenomena of a particular type or within a certain domain. Instead, the relativity theories set down principles which apply to all physical laws and restrict the ways in which they can be put together. Whether those principles are actually true is something that needs to be tested against experiment and observation, but the assumption that they do hold has far‐reaching implications for how physical laws can be constructed. In particular, the role of symmetries of Nature is highlighted, which is a defining feature of how modern physics is constructed; as such the relativity theories often give students the first glimpse of how contemporary theoretical physics is done.
Both the theories focus on how physical phenomena are viewed in different coordinate systems, with the underlying principle that the outcome of physical processes should not depend on the choice of coordinates that we use to describe them. Special Relativity restricts us to so‐called inertial frames, where the term frame means a set of coordinates to be used for describing physical laws. As we will see, this restricts us to coordinate transformations which are linear in the coordinates, corresponding to coordinate systems moving relative to one another with constant velocity, and/or rotated with respect to one another. This turns out to be a suitable framework for considering all known physical laws except for those corresponding to gravity.
Einstein's remarkable insight, leading to the General Theory of Relativity, was that allowing arbitrary non‐linear coordinate transformations would allow gravity to be incorporated. Indeed, if we want to allow non‐linear transformations, we have to include gravity. Understanding the motivations for, and implications of, this extraordinary statement is the purpose of this book. But for now, we place the focus on Special Relativity, emphasising those features that will later generalise.
2.1 The need for Special Relativity
In Newtonian dynamics, the equations are invariant under the Galilean transformation which takes us from one set of coordinates left-parenthesis t comma x comma y comma z right-parenthesis to another left-parenthesis t prime comma x prime comma y prime comma z prime right-parenthesis according to the rule
(2.1)t prime equals t semicolon x prime equals x minus v t semicolon y prime equals y semicolon z prime equals z comma
where v is the relative speed between the two coordinate systems, which have been aligned so that the velocity is entirely along the x direction. [NB primes are not derivatives!] Each coordinate frame is idealised as extending throughout space and time, providing the scaffolding that lets us locate physical processes in space and time. We introduce an event as something which happens at a specific location in space and at a specific time, such as the collision of two particles.
Typically any observer will want to choose a coordinate system to describe events, and will be located somewhere within the coordinate system. Commonly, though not always, observers will decide to choose coordinate frames that move along with them as a natural way to describe the phenomena as they see them, and so it can be useful to sometimes think of a coordinate system as being associated to a particular observer who carries the coordinate system along with them. For instance, we might consider two different observers moving at a constant velocity with respect to one another, and ask how they would describe the same physical process from their differing points of view.
When we refer to invariance of a physical quantity, we mean that a physical quantity expressed in the new coordinates is identical to the same quantity expressed in the old ones. That means that observers in relative motion agree on its value.
In particular, acceleration is invariant in Newtonian dynamics; it depends on second time derivatives of the coordinates of, for example, a moving particle, and the second time derivatives of x and of x prime are equal. An everyday example is that an object dropped in a train moving at constant velocity appears, to an observer in the carriage, to follow exactly the same trajectory as it would were the train stationary.
The Galilean transformation is characterised by a single universal time coordinate that all observers agree upon. Combining relative velocities in each of the coordinate directions means that generally x prime not-equals x, y prime not-equals y, and z prime not-equals z, but t prime always remains equal to t. The idea of a universal time sits in good agreement with our everyday experience. However, our own direct perceptions of physical laws probe only a very restrictive set of circumstances. For example, we are unaware of quantum mechanics in our day‐to‐day life, because quantum laws such as Heisenberg's Uncertainty Principle are significant only on scales far smaller than we can personally witness. Hence, we cannot immediately conclude that invariance under the Galilean transformation should apply to all physical laws.
Indeed, it was already known in Einstein's time that Maxwell's equations, describing electromagnetic phenomena including the propagation of light waves, are not consistent with Galilean invariance. For example, they state that the speed of light is independent of the motion of a source, whereas the Galilean transformation would predict that light would emerge more rapidly from a torch if its holder were running towards you. In a famous thought experiment (i.e. an experiment carried out only in the mind, not in the laboratory), Einstein tried to envisage what would happen if one tried to catch up with a light wave by matching its velocity, knowing that Maxwell's equations would not permit a stationary wave.
One possible resolution of this would be if there were special frame of reference in which Maxwell's equations were valid, a frame that came to be known as the aether. However, since the Earth revolves around the Sun, it cannot always be stationary with respect to this aether. In the late 1880s, Albert Michelson and Edward Morley sought to detect the motion of the Earth relative to this aether, using an interferometer experiment. It should have had the sensitivity to easily see the effect, given the known properties of the Earth's orbit, yet no signal was found, putting the existence of the aether in doubt.
From the viewpoint of wanting a unified view of physical laws, it makes little sense that different types of physical laws should respect different invariance properties. After all, electromagnetic phenomena lead to dynamical motions. This incompatibility posed a stark problem for physics.
Einstein's 1905 paper resolved this seeming paradox decisively in favour of electromagnetism. Based on his thought experiments, he demanded that physical laws satisfied two postulates:
1. The laws of physicsare the same in all inertial frames.2. The speed of light, denoted c,is the same in all inertial frames and independent of the motion of the source.
As remarked above, inertial frames are those which move with a constant velocity with respect to one another. The requirement that the laws of physics be the same in each is inherited from the Galilean transformation, which also requires it. Another way of expressing this first postulate is to say that there is no possible experiment an observer can carry out to measure their absolute velocity.
But the second postulate then requires that the coordinate transformation between frames must mix space and time, as we are about to see. It is inconsistent with the notion of a universal time coordinate, and requires that invariance under the Galilean transformation be abandoned. If Nature's laws are to be invariant under coordinate transformations, the invariance must be of another type.
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