Relativity in curved spacetime : life without special relativity

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Einstein demanded something observable to make the difference. When we try to accelerate, we feel inertial forces. These are the forces that make us dizzy when we spin in a fun fair; or they are the forces that throw our coffee in the air when our airplane hits an air pocket.

General relativity - Wikipedia

These forces, Einstein understood Mach to assert, arise from an interaction between the mass of our body and our coffee and all the other masses of the universe, distributed in the stars. Einstein first called this idea the "relativity of inertia" and later, in , "Mach's Principle. In the case of Einstein's two fluid spheres , the bulge of one of them would now be explained by the fact that this bulging sphere was rotating with respect to all the other masses of the universe, whereas the other sphere was not.

That would be the observable difference between the two fluid bodies. This analysis was clearly inspired by Mach's famous account of Newton's bucket experiment. Newton had noted that water in a spinning bucket adopts a concave surface. The concavity is a result, Newton urged, of its rotation with respect to absolute space. No, Mach had responded several hundred years later, all one has in the case of Newton's bucket in rotation with respect to the stars. We cannot know anything more than what our direct observations tell us. All they tell us is that these inertial forces arise when we accelerate relative to the stars.

The weakness of this analysis is that there is no account of how rotation with respect to distant masses could produce these inertial forces. In , Einstein hoped that his emerging theory of gravity would provide the mechanism. It could then satisfy Mach's Principle and, through it, generalize the principle of relativity to acceleration. For in a theory that satisfies Mach's Principle, no state of motion is intrinsically inertial or accelerating.

When we see something accelerating, it is not accelerating absolutely in such a theory; it is merely accelerating with respect to the stars. Preferred inertial motions need not enter into the account any more. All motion, accelerated or inertial, would be relative. To deliver this sort of account of inertial forces, Einstein's theory would need to break down the strict division between inertial and accelerated motion of his special theory of relativity.

The principle of equivalence promised to weaken this division. According to it, whether the physicist in the box was to be judged accelerating or not depended on your point of view. An inertial observer would judge the physicist to be accelerating uniformly in a gravitation free space. The physicist would judge him or herself to be unaccelerated in a gravitational field.

It was a first step towards generalizing the principle of relativity to acceleration, Einstein believed. By his own later judgment, Einstein did not, in the end, find a theory that fully satisfied Mach's Principle. The immediate benefit of his new principle of equivalence, however, was that it let Einstein learn a lot about gravitation. For the principle delivered to Einstein one special case of a gravitational field that, he believed, conformed with relativity theory and in which all bodies truly fell alike.

Einstein's program of research on gravity in the five years following was simply to examine the properties of this one special case and to try to generalize them to recover a full theory. His early hope was that the generalization of the principle of relativity would somehow emerge in the course of those investigations. Two properties of this special case of the gravitational field were noteworthy. First, Einstein recognized that clocks run at different rates in the box of his thought experiment according to their location. A clock placed lower in the created field runs slower.

Einstein immediately generalized that effect to all gravitational fields. Clocks deeper in a gravitational field run slower. A clock in the sun would run slower than one on earth--if only we could have a clock in the sun without it being destroyed by the heat of the sun. It turns out we can find clocks in the sun.

Radiating atoms radiate in very definite frequencies of light according to which element they are. That means that they behave like little clocks. Their running slower is manifested in a slight reddening of the light they emit. Einstein computed an effect on the wavelength of sunlight of one part in two million. While Einstein did not use spacetime diagrams in , they provide an easy way to see that clocks run at different rates according to their position when they accelerate in a Minkowski spacetime.

The effect is driven almost entirely by the relativity of simultaneity. Now consider an observer who accelerates with the rightmost "B" clock, that is, the clock higher up in the created field. As the clock changes speed, that observer's hypersurfaces of simultaneity will tilt so that the B observer will judge the A clock to be lagging successively more behind. When B's clock reads 2, B will judge the A clock to read 1; when B's clock reads 4, B will judge the A clock to read 2. Overall, B will judge A's clock to be running at half the B clock's speed. The effect, the figure shows, is entirely due to the relativity of simultaneity.

The geometry of uniform acceleration in a Minkowski spacetime turns out to be especially simple. The hypersurfaces of simultaneity of an observer accelerating with the B clock turn out to coincide with the hypersurfaces of simultaneity of an observer accelerating with the A clock. Hence the observer moving with clock A will agree that the A clock is running slower and the B clock faster.

Life without special relativity

When the A observer's clock reads 1, A will judge B's clock to read 2. When the A observer's clock reads 2, A will judge B's clock to read 4. The second important effect pertained to light. An unaccelerated observer finds that light propagates in a straight line in the space of special relativity. Here, for example, is such a light flash propagating across the box of Einstein's thought experiment.


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For the physicist accelerating with the box, however, the light will be judged to fall, just like everything else in the box. As a result, the physicist will find the light's path to be bent downward by the gravitational field. Einstein generalized this result to arbitrary gravitational fields. This generalization enabled him to make one of the most celebrated predictions of his theory. A ray of starlight grazing the sun would be bent as the light fell into the sun's gravitational field.

This bending would be manifested as a displacement of the star's apparent position in the sky and this displacement would be visible at the time of a solar eclipse. In , Einstein had predicted the gravitational bending of light. But he did not realize that it might actually be tested at the time of a solar eclipse. After his Jahrbuch article, Einstein's efforts were redirected towards the puzzle of the quantum. In , however, he returned to theorize about gravity. He realized then that his prediction of the gravitational bending of light could be tested at a solar eclipse.

He wrote another paper developing this idea and also other aspects of his theory. Einstein was keen to see this test undertaken. The greatest difficulty was that it required a solar eclipse and that meant that astronomers must place themselves precisely in its path. In , Einstein wrote to the American astronomer G. Hale asking whether the test could be undertaken without an eclipse. Hale responded that it could not.

The brightness of the sky near an uneclipsed sun is just too great. In August , there was a promising eclipse of the sun that would be visible from the Crimea. Einstein's colleague, the astronomer Erwin Freundlich, mounted an expedition to the Crimea to observe and photograph the eclipse. Unfortunately for Freundlich, the First World War broke out.

Since he was German, a citizen of an enemy nation, the Russians interned him and confiscated his equipment. Fortunately for Einstein, no measurement was taken. Einstein's theory of was not yet the complete general theory of relativity. In his earlier theory, there was no curvature of ordinary space in the vicinity of the sun. As a result, as we saw in another chapter , his theory predicted the same deflection as Newtonian gravitation theory assuming light consists of massive corpuscles. It was half the deflection predicted by the final theory. Had the test been carried out successfully, it would have produced a result that contradicted Einstein's earlier theory.

In , Einstein had also concluded that the speed of light , and not just its direction, would be affected by the gravitational field. The effect was closely connected with the gravitational slowing of clocks and is almost entirely a consequence of the relativity of simultaneity. One can see how it comes about with a similar set of spacetime diagrams.

A light signal propagates from A to A' and a second light signal propagates from B to B'. The figure shows the hypersurfaces of simultaneity of an inertial observer. Of course the inertial observer will judge the two light signals to propagate at the same speed. That is just familiar special relativity. We notice also that, initially, the four clocks A, A', B, B' run in synchrony according to the judgments of simultaneity of the inertial observer.

Hence using the readings of these clocks directly, we will infer that the two light signals propagate at the same speed. In more detail, we note that the distance from A to A' equals the distance from B to B'; and each light signal takes the same time to traverse the distance. Both light signals leave when the local clocks read 0 and arrive when the local clocks read 3. Hence using these local clock readings, we infer that the two light signals travel at the same speed. Now consider how these processes are judged by an observer who accelerates with the clocks.

All that changes in the analysis that follows is that we use different judgments of simultaneity. That leads to the judgment of differing speeds for the propagation of light.

Let us take the observer who accelerates with clock B. That observer's hypersurfaces of simultaneity will tilt more and more as clock B gains speed from the acceleration. This was the effect that led observer B to judge that the A clock was running slower than the B clock. This same tilting will lead observer B to judge that the AA' light signal propagates at roughly half the speed of the BB' light signal. Both signals traverse the same distance. Recall that the judgments of simultaneity of accelerating observers who move with the clocks agree, since they agree on the hypersurfaces of simultaneity.

So we can choose any one of the accelerating observers and get the same outcome.

Einstein's Pathway to General Relativity

Each of the accelerating observers will judge the transit time for BB' to be roughly half that of AA'. They will agree that light propagates slower on the left side of the figure, that is, deeper in the created field. Applying the principle of equivalence , we now conclude that the same slowing manifests in a gravitational field. A light signal deeper in the gravitational field at A propagates slower than a light signal higher in the gravitational field at B. The conclusion that gravity slows the speed of light caused Einstein some trouble with unkind contemporary critics. Einstein had first based his theory of of the striking idea of the constancy of the speed of light, but he now seemed to be retracting it.

By , Einstein had developed all these ideas into a fairly complete theory of static gravitational fields, that is gravitational fields that do not vary with time and admit well defined spaces. The most striking characteristic of the theory was that the intensity of the gravitation field , the gravitational potential, was given by the speed of light. So as one moved to different parts of space, the intensity of the gravitational field would vary in concert with the changes in the speed of light. As late as , some five years after Minkowski's work, Einstein was loath to use spacetime methods.

While I have developed the clock slowing and light slowing effects using spacetime diagrams, Einstein did not do this. His method of analysis was algebraic. He represented the processes by equations in which speeds and times appeared as variables. He rarely if ever drew diagrams such as given above. What Einstein now needed was a way to extend these results to the more general case of gravitational fields that vary with time.

That, it turned out, required Einstein to move well beyond the mathematics he knew. Another thought experiment pointed the way. If one has a circular disk at rest in some inertial reference system in special relativity, the geometry of its surface is Euclidean.

It will be useful to spell out what that means in terms of the outcomes of measuring operations. If the disk is ten feet in diameter, then it means that we can lay 10 foot long rulers across a diameter. That means that we can traverse the full circumference of the disk by laying 31 rulers around the outer rim of the disk. Thus we measure the circumference of the rotating disk to be greater than 31 feet, the Euclidean value. In other words, we find that the geometry of the disk is not Euclidean.

The significance of this thought experiment was great for Einstein. Through his principle of equivalence, Einstein had found that linear acceleration produces a gravitational field. Now he found that another sort of acceleration, rotation, produces geometry that is not Euclidean.

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Einstein had all this in place by the summer of He knew that gravitation could bend light and slow clocks. He expected that the final theory would somehow involve accelerations in a new way and that such accelerations came with a breakdown of Euclidean geometry. He also knew that the natural arena in which to conduct relativity theory is Minkowski's spacetime. We now call it "tensor calculus. This, in Einstein's view, was just the mathematical instrument needed, for Einstein had associated an extension of the principle of relativity to acceleration with an expansion of the spacetime coordinate systems of a theory.

Einstein took a series of bad turnings and ended up with the wrong gravitational field equations--not the celebrated Einstein equations that appear in all the modern textbooks. He needed over two years of painful work first to recognize that something had gone wrong and then to find the final equations. The precise causes that brought about these wrong turnings remain a point of debate in the history of general relativity literature. We can identify two elements, however, that played a role in misleading Einstein.

Second, Einstein used a different style of theorizing to the one largely used in these chapters.


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Here, we have used a geometrical approach, emphasizing the picturing of gravitational effects in geometric diagrams. Einstein, however, labeled events in spacetime with arbitrarily coordinate numbers and expressed all his results in terms of equations relating these coordinates. Einstein knew that this labeling of spacetime events with coordinates was purely arbitrary and that all his results had to be independent of the particular coordinate system used.

However knowing this in the abstract and carrying through the demand in all details are two different things. Einstein's "Entwurf" field equations applied only to specialized sets of coordinate systems. He called them "adapted. One of the low points in his struggle with coordinate systems came when Einstein used an ingenious argument--the " hole argument "--to show that gravitational field equations like the ones of his final theory are inadmissible on physical grounds.

While the hole argument did not warrant that conclusion, it has been rehabilitated in recent work in philosophy of space of time, where it now lives a good life. See, " The Hole Argument. Here's one page on which Einstein writes down the Riemann curvature tensor for the first time and finds it hard to see how it can be used in his gravitational field equations. The notebook has many wonderful insights into the detailed steps of Einstein's work; and there are still pages whose content eludes us. Each is a little puzzle for us solve. One remarkable page that defeated me was solved by Tilman Sauer.

It will not be at all obvious what Einstein is computing on the right hand side of this page. At least it was not to us for a long time. So let me tell you what Einstein computes. The result is so close to the central idea of Einstein's general theory of relativity that one has to look closely to see how the two differ:. With the publication of the "Entwurf" paper in mid , Einstein felt that the major work on the theory was done.

Only the details needed to be clarified. The feeling did not last. As the months and years passed. Einstein worked harder and harder to convince himself that all was well with a theory that was misshapen in its fundamental equations. In the summer and fall of , the clues that his old theory was wrong mounted. He knew that his theory of did not accommodate the anomalous motion of Mercury.

He then found that it did not extend the relativity of motion to rotation. Finally, he found that an improved and much more sophisticated development of the theory of late did not demonstrate its uniqueness, as he then believed. In mounting despair and desperation to save his theory, he returned to the thinking of and He was mistaken, he saw, to restrict his theory only to the special "adapted" coordinate systems of the "Entwurf" theory. He needed a theory that would work in as many coordinate systems as possible. Ideally it would work in all coordinate systems.

That is, it would be "generally covariant. What followed was the most exhausting and exhilarating month of Einstein's life. He sent the Prussian academy bulletins on his reformulated theory, one per week, each correcting errors and incompleteness of the previous bulletins.

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What no doubt lent special urgency to this extraordinary behavior was the fact that David Hilbert , the greatest mathematician of the era, had also become interested in the theory and had started the project of rewriting the gravitational field equations of Einstein's old "Entwurf" theory in a mathematically more elegant formulation.

The first communication was sent on November 4 and was written as it if was the last. It was not. The following week, November 11 , Einstein sent a correction. And the following week, November 18 , a jubilant Einstein communicated the remarkable news that his newly formed theory accounted for the anomalous motion of Mercury. The field equations of the theory were still not quite right. It was only in the fourth communication of November 25 that the final equations appeared.

Here are those equations as he published them for the first time on 25 November Here he writes them later in the simple case of a matter free spacetime:. Copyright John D. Minor edits February 26, More December 29, Classical physics General relativity A mass moves freely in space, except that it is constrained to a two-dimensional surface in the three-dimensional space. A mass moves freely in spacetime. That is, it is in free fall, so that gravity acts on it through the curvature of spacetime.

Its spatial trajectory is a geodesic of the two-dimensional surface. That is, it traces of curve of shortest length in the surface. Its spacetime trajectory is a geodesic of the spacetime. That is, it traces a curve of extremal spacetime interval in spacetime. The seven years of work divides loosely into two phases.

The earlier phase of his work from to was governed by powerful physical intuitions that seemed as much rationally as instinctively based. Einstein felt a compelling need to generalize the principle of relativity from inertial motion to accelerated motion. He was transfixed by the ability of acceleration to mimic gravity and by the idea that inertia is a gravitational effect. These ideas finally issued in a theory of static gravitational fields in In it, gravity bends light and slows clocks and the speed of light varies from place to place. The major transition to general relativity came after the summer of and into early Einstein struggled to incorporate these ideas into a more general physical theory.

He was drawn to use the mathematics of curvature as a means of formulating the new theory. To learn the mathematics, he collaborated with his mathematician friend, Marcel Grossmann. Together, they produced the first draft of the general theory of relativity. The remaining years, to , were devoted to the labored work of correcting and perfecting his draft of As the mathematics of curvature took a more controlling position in the later phases, Einstein's work began to change.

The theorizing was governed increasingly by notions of mathematical simplicity and naturalness. When the theory was completed, Einstein's starting point was quite distant.

Einstein's Relativity and Everyday Life

It is impractical in this chapter to review all these considerations. Einstein's intricate mathematical struggles in the later years cannot easily be described in informal terms. However some of his earlier physical reflections are so famous and so characteristic of Einstein , that they must be mentioned. Finally a caution to those seeking to learn general relativity from these pages.

The considerations below are the ones that guided Einstein towards general relativity.

Whether these guides persist as founding principles of general relativity remains debated today. In particular, there is increasing consensus that the final theory does not extend the principle of relativity to accelerating motion in any interesting way. You should treat what follows as interesting reports on Einstein's intellectual biography. You may well find it hard to connect some of the ideas to be laid out below with the final theory. This effect came about from an apparently accidental agreement of two quantities in Newtonian theory: the inertial mass of a body happens to equal its gravitational mass exactly.

Einstein now believed that this equality could be no accident. He needed to find a gravitation theory in which this equality is a necessity. The inertial mass of a body measures its resistance to acceleration when a force is applied to it. The gravitation mass of body measures how it responds to a gravitational field. For more, see this. Einstein later complained about this version of the principle, objecting that one could not in general transform away an arbitrary gravitational field over an extended region of space. His original formulation and the one to which he adhered for his entire life proceeded differently.

He turned around the original idea of free fall eradicating gravitation. Acceleration can also produce a gravitational field. From his geometrical studies, he believed the gravitational field was simply a manifestation of curved space-time. Hence, Einstein could show that accelerating frames were represented by non-Euclidean space. The third key step for Einstein involved resolving complications that had arisen when special relativity was applied to Newtonian gravitational physics. Put simply: Newtonian gravity predicted that if the sun was removed from the centre of the solar system, the gravitational effect on the Earth would be instantaneous.

However, special relativity says that even the gravitational effect of the sun disappearing ought to travel at the speed of light. So mass clearly determined the strength of the gravitational field. Special relativity tells us that mass is equivalent to energy, so the energy—momentum density must also determine the gravitational force. Rotating frames non-inertial imply non-Euclidean or curved space-time. The equivalence principle asserts that accelerating frames i.

From special relativity, mass is equal to energy, and from Newtonian physics, mass is proportional to strength of gravity. Hence, Einstein was able to conclude that energy-momentum density causes, and is proportional to, space-time curvature. From to , Einstein published several papers as he laboured to complete the general theory.

Some of these papers contained errors and took Einstein down theoretical pathways that ultimately were not productive. But the final result, that the energy—momentum density of matter curves space-time, like a bowling ball curves a flat sheet of rubber, and that the motion of a mass in a gravitational field is based on the curvature of space time, just as a bowling ball moves freely on a curved rubber sheet, is surely one of the greatest insights of human intellect.

It is possible we would have had to wait many decades. However, in the cat would have been out of the bag. This astonishing discovery could only have made sense if space-time was curved. Maybe it still should. Screen music and the question of originality - Miguel Mera — London, Islington.

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