Spacetime

11 Key excerpts on "Spacetime"

• 

  eBook - PDF

  Gravity from the Ground Up

  An Introductory Guide to Gravity and General Relativity

  • Bernard Schutz(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  Einstein showed that we cannot separate space from time easily. We learn the language of Spacetime and illustrate the entanglement of space and time with an example loosely based on the legend of William Tell. We will see how to describe the geometry even of special relativity. world is not constructed in the way we may have thought, and certainly not in the way that Galileo and Newton thought. Time is not absolute: different experimenters measure it differently, and no single experimenter has a better definition than an- other. Nor is space absolute: solid objects have different lengths when measured by experimenters moving at different speeds. These ideas were just as troubling and counter-intuitive to physicists of Ein- stein’s day as they are today to new students who encounter them for the first time. When physicists began to think more deeply about why the ideas were troubling, they found that it helped them to stop thinking about time and space as distinct and separate things, and instead to join them together. Since time is one-dimensional (the history of ancient Rome, for instance, can be ordered along a single line) and space has three dimensions, their combination is a four-dimensional realm. We call this Spacetime. A single point of Spacetime occupies, therefore, both a particular location in space and a particular moment in time. Just as space is the collection of all “places”, or points, Spacetime is the collection of all “happenings”, or events. A Spacetime diagram is a graph that records the entire history of an experiment or of some other process. We can clarify what this means by drawing a Spacetime diagram, as in Fig- ure 17.1. I will illustrate the idea by recording in this diagram, in a simplified way, the history of the legendary episode where the Swiss patriot William Tell was com- pelled to shoot an apple from the head of his son.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  The Fourth Dimension

  Enigma of Time

  • Dayalanand Roy, Ph.D.(Authors)
  • 2021(Publication Date)
  • BrownWalker Press

    (Publisher)

  43

  The Concept of Time in Physics

  What is the idea of time in physics? According to Einstein, the time of an event is the reading given by a clock placed near the event whose construction is similar to other clocks. “… A time value is associated with every event which is essentially capable of observation,”44 suggests Einstein. However, the concept of time in physics has multiple feathers in its cap. H. Dieter Zeh has thoroughly described various physical aspects of time in his work, The Physical Basis of the Direction of Time .45

  Though the above definition serves well the purpose of defining the ‘time’ of an event as recorded by a physicist, yet it may not satisfy the curiosity of a common man when he asks- “What do we mean by the ‘time’ being recorded by a clock?” and we answer- “Time is what a clock reads”. To me, this seems to be a circular argument. This is just like the argument that when asked what length is, we reply length is what a meter rod measures. And what does a meter rod measure? It measures length. No one will agree that length is properly explained from this argument. A better explanation is- length is one of the spatial dimensions of an object.

  Physics gives another description of time- ‘time is the fourth dimension of space’. It was further developed and modified by Einstein when he gave his Special Theory of Relativity- he fused this fourth dimension with the other three dimensions of space.

  However, when a common man hears that time is the fourth dimension of space, he is again perplexed by this idea. He can imagine, though roughly, what space is. When he looks towards sky, he thinks he is looking at space. He thinks that whatever he sees around him is located in this space. But where in space is this fourth dimension- the dimension of time? We have seen above that though initially Einstein believed in the independent existence of space, after bringing his grand General Theory of Relativity, his concept was modified. “… Space-time does not claim existence on its own, but only as a structural quality of field”40

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  The World According to Physics

  • Jim Al-Khalili(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  CHAPTER 3 SPACE AND TIME In such a short book I am unable to cover all areas of physics, fascinating though so many of them are. Instead, I have distilled our current understanding of the physical universe down to three central pillars: three pictures of reality that come from very different directions. The first of these, introduced in this chapter and the next, is built on the work of Albert Einstein in the early twentieth century. It lays out our present understanding of the way matter and energy behave within space and time on the very largest scales due to the influence of gravity—an understanding that is encompassed in his famous general theory of relativity. In order to paint Einstein’s picture of the world, we must start with the canvas itself. Space and time are the substrates in which all events take place. However, such concepts are slippery. Common sense tells us that space and time should be in place from the start—that space is where events happen and the laws of physics are acted out, while the inexorable passage of time is, well, just is. But, is our commonsense view of space and time right? An important lesson physicists must learn is to not always trust common sense. After all, common sense tells us that the Earth is flat, but even the Ancient Greeks understood that its sheer size meant we could not easily discern its curvature, but that there were simple experiments they could perform to prove that it was in fact a sphere. Similarly, everyday experience tells us that light has the properties of a wave and therefore cannot also behave as though it were made up of a stream of individual particles. If it were, how could we explain interference patterns? And yet it has been proven beyond doubt, through careful experiments, that our senses can deceive us when it comes to the nature of light

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Einstein Relatively Simple

  Our Universe Revealed in Everyday Language

  • Ira Mark Egdall(Author)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  Minkowski declared that Einstein’s theory of special relativity is best visualized in a four-dimensional “space-time” continuum. What are these four dimensions? They are the three space dimensions ( up-down, left-right, and forward-back) and the time dimension. Minkowski proposed the revolutionary idea that time and space are not separate entities, but joined together mathematically.

  Minkowski proposed his concept as an alternate to Einstein’s theory rather than as a mathematical adjustment. Einstein was not amused. Minkowski’s paper contained a single physical prediction. Einstein was quick to point out it “failed to account for a known phenomenon.”a , 11

  Minkowski in turn had said of Einstein; “The mathematical education of the young physicist [Albert Einstein] was not very solid, which I am in a good position to evaluate since he obtained it from me in Zurich some time ago.”12 The rivalry between Einstein and Minkowski “was carried on at a level that few recognized.”13 The unpleasantness ended with Minkowski’s premature death of appendicitis in 1909 — he was 44 years old.

  Wherewhen

  What exactly does the term Spacetime mean? To help visualize this, let’s first look at a plot of length versus width (Fig. 8.2(a)). Here the area between the two axes represents space.

  Now let’s construct a Spacetime diagram (Fig. 8.2(b)). What is a Spacetime diagram? It is simply a plot where one axis is time and the other axes are space. (We only plot one of the three space dimension here for simplicity.) So space is the horizontal axis and time is the vertical axis. What does the area between the axes of space and time represent? It represents Spacetime.14

  Are we in Spacetime right now? Yes, most definitely. But our common way of thinking is not conducive to the unification of space and time. We have to think differently — not in terms of places in space or happenings in time — but in terms of events which have a location in both space and time, i.e., in Spacetime.

  Strictly speaking, an event is something that occurs at a single point in space and at a single point in time. Thus an event is a single point in Spacetime. In addition, the very language we use to talk about events needs modification. For example, if you are asked where you were born, you reply with a location in space, such as in San Diego, California. If you are asked when

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Universe And The Atom, The

  • Don B Lichtenberg(Author)
  • 2007(Publication Date)
  • World Scientific

    (Publisher)

  Some people have even speculated about reasons for this fact, but nobody has yet come up with a satisfactory answer. Some people have speculated that there are additional dimensions, but that these 36 CHAPTER 3. NEWTON’S IDEAS ABOUT SPACE AND TIME are in directions in which space is curved, and that the extra dimen-sions curl around in such a small distance that we are unaware of them. However, at present no one has demonstrated the existence of extra spatial dimensions in our universe. In physics the concept of time seems just as basic as the concept of space. Time is commonly thought of as a scalar, because only one number is required for its specification. For example, a family may sit down to dinner at 7 p.m. This time is understood because we have standard units of time (hours, minutes, seconds, to name a few), and standard time zones. Just as we have standard rulers to measure distance, we have standard clocks to measure time. But there is a sense in which time is not a scalar. Suppose you are asked to describe where and when you saw a woodpecker on a pole. You might respond by giving four numbers: 30 meters north, 40 meters west, 10 meters up, 4:20 p.m. These numbers might be considered the components of a 4-vector. It is not the usual vector, to be sure, because three components specify distances and one com-ponent specifies a time. Thus, 4-vectors are not vectors in ordinary space but vectors in Spacetime. It is in this sense that time is some-times called the fourth dimension. There is an apparent difficulty with this interpretation, because time has different units than dis-tance. However, because the product of speed and time has the same units as distance, we can overcome the problem of units by multi-plying time by a speed, usually the speed of light. Then the fourth number is the distance traveled by a beam of light in the time in question. Einstein’s special and general theories of relativity (discussed in

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Something Deeply Hidden

  Quantum Worlds and the Emergence of Spacetime

  • Sean Carroll(Author)
  • 2019(Publication Date)
  • Oneworld Publications

    (Publisher)

  general relativity, his theory of gravity and curved Spacetime. The crucial insight was that four-dimensional Spacetime isn’t just a static background on which the interesting parts of physics take place; it has a life of its own. Spacetime can bend and warp, and does so in response to the presence of matter and energy. We grow up learning about the flat geometry described by Euclid, in which initially parallel lines remain parallel forever and the angles inside a triangle always add up to 180 degrees. Spacetime, Einstein realized, has a non-Euclidean geometry, in which these venerable facts are no longer the case. Initially parallel rays of light, for example, can be focused together while moving through empty space. The effects of this warping of geometry are what we recognize as “gravity.” General relativity came with numerous mind-stretching consequences, such as the expansion of the universe and the existence of black holes, though it has taken physicists a long time to appreciate what those consequences are.

  Special relativity is a framework, but general relativity is a specific theory. Just like Newton’s laws govern the evolution of a classical system or the Schrödinger equation governs the evolution of a quantum wave function, Einstein derived an equation that governs the curvature of Spacetime. As with Schrödinger’s equation, it’s fun to actually see Einstein’s equation written out, even if we don’t bother with all the details:

  Rμν —(½)Rgμν = 8πGTμν

  The maths behind Einstein’s equation is formidable, but the basic idea is simple, and was pithily summarized by John Wheeler: matter tells Spacetime how to curve, and Spacetime tells matter how to move. The left-hand side measures the curvature of Spacetime, while the right-hand side characterizes energy-like quantities, including momentum, pressure, and mass.

  General relativity is classical. The geometry of Spacetime is unique, evolves deterministically, and can in principle be measured to arbitrary precision without disturbing it. Once quantum mechanics came along, it was perfectly natural to try to “quantize” general relativity, obtaining a quantum theory of gravity. Easier said than done. What makes relativity special is that it’s a theory of Spacetime rather than a theory of stuff within Spacetime. Other quantum theories describe wave functions that assign probabilities to observing things at definite, well-defined locations in space and moments in time. Quantum gravity, by contrast, will have to be a quantum theory of Spacetime itself. That raises some issues.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Principles of Cosmology and Gravitation

  • Michael V Berry(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  4 Curved Spacetime and the physical mathematics of general relativity 4.1  Particle paths and the separation between events Now we must start to get to grips with general relativity itself. We wish to predict the motion of a body under the action of gravitation, avoiding the difficulties already described which result from the Newtonian formalism. Therefore we employ a Spacetime description, and seek the world line of the body, that is, the locus of successive events in its history. Each event is a point in Spacetime, specified by four coordinates x i, where i = 0, 1, 2, 3. It is conventional to take x 0 as a time coordinate t, and x 1, x 2 and x 3 as three coordinates giving the spatial position r (= x, y, z; r, θ, ϕ etc.). We have in mind here a lattice formed by rigid ‘standard rods’. whose intersections give spatial positions; at each intersection is a ‘standard clock’which can be synchronised with all the others by means of a technique involving light signals – it is not necessary for us to go into details here. For any event, the nearest intersection gives r, and the clock reading there gives t. However, much more general specifications of events can be employed. For example, an explosion in the atmosphere can be specified by taking the readings of clocks carried by four randomly-moving aeroplanes, at the moment the explosion is seen or heard; the clocks need not be synchronised and may even run at different speeds, but they must keep going. The four readings thus obtained specify the explosion event uniquely, and form a perfectly acceptable set of coordinates x i. This freedom in choosing coordinate systems is particularly useful in view of the fact that a rigid body, such as a ‘standard rod’, is not easy to define in relativity (the trouble arises because the speed of sound in such a body would be infinite, and therefore greater than that of light)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Time: A Philosophical Introduction

  • James Harrington(Author)
  • 2015(Publication Date)
  • Bloomsbury Academic

    (Publisher)

  This is equivalent to postulating a true Newtonian absolute rest, but it gives a natural sense to the principle of inertia according to which free bodies follow straight paths. Alternatively, Galilean space-time only has spatial distances defined between events at the same time. This requires an additional structure to specify straight paths for inertial motion. For more details, see John Earman’s World Enough and Space-time in the Further Reading section. Time as a Physical Quantity 181 This is no longer true once we move to special relativity and Minkowski space-time. This is because the corresponding invariant is no longer a distance but a velocity, the velocity of light. As we saw above, Einstein demonstrated that jointly satisfying the principle of relativity and the constancy of light requires that space and time transform differently than they would if Galilean relativity held. Just as the Galilean transformations represent the fundamental physical facts of classical physics—spatial separation, temporal interval, and acceleration—as equivalent in each reference frame, so the Lorentz transformations do the same for Einsteinian physics. In their simplest form, for systems of coordinates X x y z t X = ¢ = { , , , } { , , , } and x h z t where the origin of the X ʹ system is moving with velocity v relative to the X -system along the x -axis of that system, they are given in Figure 6.9. Now let’s consider the speed of a light pulse emitted from the joint origins of two such coordinate systems at time zero in both frames. Einstein’s postulate states that the speed of light in each frame must be the same, as represented in Equation 6.7: x b u = -( ) x t (6.2) t b u = -( / ) t x c 2 (6.3) h = y (6.4) z = z ; (6.5) b u = -1 1 2 / ( / ) c (6.6) Figure 6.9 Lorentz transformations for a system moving with the velocity, v .

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introducing Relativity

  A Graphic Guide

  • Bruce Bassett, Ralph Edney(Authors)
  • 2014(Publication Date)
  • Icon Books

    (Publisher)

  HOW COULD WE MAKE A SPACE WHICH WOULD ALLOW US TO RECORD THE POSITION OF EVERY SINGLE ATOM AT EVERY MOMENT IN THE UNIVERSE’S HISTORY?SO EACH ATOM REQUIRES ITS OWN 4-DIMENSIONAL SPACE...WELL, FOR EACH ATOM WE NEED FOUR NUMBERS, ITS POSITION IN SPACE – THERE NUMBERS – AND ITS TIME – ONE NUMBER...... YES, AND 5 ATOMS REQUIRE A 4×5 = 20-DIMENSIONAL SPACE

  Slicing Spacetime

  However, we have an infinite number of atoms in this example, so the complete space is infinite dimensional (4 × infinity = infinity). We need an infinite number of numbers to record uniquely where all the atoms are. This space, though we will not need it, is known as configuration space in mechanics, since it gives the configuration of the system.

  Notice that by thinking of spaces abstractly, one gives up the need to be able to visualize them in everyday terms.

  A POWERFUL WAY IS TO CONSIDER “SLICES” OF THEMBUT IT IS OFTEN USEFUL TO TRY TO VISUALIZE THEM

  We can slice up Spacetime (which is four-dimensional) into three-dimensional slices which we can visualize.

  How to View Spacetime

  One of the big advantages of abstraction by letting go of the need to visualize things as they would look in our world is that we can give up the urge to think continually of spaces in terms of them being inside bigger spaces.

  FOR EXAMPLE, WE TYPICALLY THINK OF A SHEET OF PAPER AS A TWO-DIMENSIONAL SPACE LYING IN OUR THREE-DIMENSIONAL SPACESO WHEN PEOPLE HEAR THAT THE UNIVERSE IS EXPANDING, THEY NATURALLY ASK...... EXPANDING INTO WHAT?

  This is a natural question, from the standard point of view, but not from our new view in which we think of a space as existing completely separately from any other space. Hence, cosmologists usually think of the expansion of the universe only as a property of Spacetime itself, namely that the distance between any points IN THE Spacetime is increasing.

  Simultaneity is Relative

  One of the key ingredients of relativity is that – unlike Newton’s view of gravity – space and time are unified into a four-dimensional space which can be sliced, like a loaf of bread, in different ways, to give “space” and “time”. But there is NO UNIQUE or preferred way to slice Spacetime. This is in fact a geometrical

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Gravity

  Newtonian, Post-Newtonian, Relativistic

  • Eric Poisson, Clifford M. Will(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Curved Spacetime The relativistic formulation of the laws of physics developed in Chapter 4 excluded gravita- tion, and our task in this chapter is to complete the story by incorporating this all-important interaction (our personal favorite!). In Sec. 5.1 we explain why relativistic gravitation must be thought of as a theory of curved Spacetime. In Sec. 5.2 we develop the elementary aspects of differential geometry that are required in a study of curved Spacetime, and in Sec. 5.3 we show how the special-relativistic form of the laws of physics can be generalized to incorporate gravitation in a curved-Spacetime formulation. We describe the Einstein field equations in Sec. 5.4, and in Sec. 5.5 we show how to solve them in the restricted context of small deviations from flat Spacetime. We conclude in Sec. 5.6 with a description of spherical bodies in hydrostatic equilibrium, featuring the most famous (and historically the first) exact solution to the Einstein field equations; this is the Schwarzschild metric, which describes the vacuum exterior of any spherical distribution of matter (including a black hole). 5.1 Gravitation as curved Spacetime 5.1.1 Principle of equivalence Relativistic gravity The relativistic Euler equation (4.59), unlike its Newtonian version of Eq. (1.23), does not contain a term that describes a gravitational force acting on the fluid. To insert such a term requires an understanding of how the Newtonian theory of gravitation can be generalized to a relativistic setting. It is tempting to attempt such a generalization by simply replacing the Poisson equation ∇ 2 U = −4π Gρ with a Lorentz-invariant generalization such as ✷U = 4π GT μ μ /c 2 = −4π G(ρ + /c 2 − 3 p/c 2 ), and replacing the term ∂ j U in the Newtonian Euler equation by something like P αβ ∂ β U .

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  The Philosophy of Space and Time

  • Hans Reichenbach(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  The assertion that measuring rods, clocks, and light rays behave according to the relations of congruence of the indefinite metric represents the geometrical formulation of the light- and matter-axioms.

  We have previously discussed (page 160) the assertion concerning the union of space and time. On the basis of the geometrical representation we can now clarify this assertion. Surely, the graphical representation of time, the combination of space and time into one manifold, is not new, since it also holds in the classical theory of time. The new content can be summarized in the following two assertions.

  First, it is maintained that the element of the manifold determined by two point-events, namely the interval, finds its natural realization by clocks, measuring rods and light rays. This means that these measuring instruments introduce into the manifold certain congruence relations of a very specific kind. It is this fact which has made the four-dimensional treatment of space and time so fruitful, and which is expressed in the statement that clocks, measuring rods, and light rays assume for the four-dimensional space-time manifold a function which is similar to the function performed by rigid rods in three-dimensional space. It is true that the classical space-time theory could have treated space and time as a four-dimensional manifold; it would even have been possible to define some metric within this manifold. However, there would have existed no physical objects that would have realized the congruence relations of this metric. The assertion that there exists a natural metric for the space-time manifold has therefore great significance for physics. In this sense we may speak of a union of space and time. This does not mean, however, that space and time lose their specific individual differences, for, clearly, clocks and measuring rods are quite different types of measuring instruments. This union of space and time, therefore, preserves their specific properties.

  Sign up to read

  Learn more about book