Flat and Curved Space-Times

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But all of them are true, or at least plausible. What gives?

What Did Einstein Mean By 'Curved' Spacetime? - Newsy

Leaving aside the issue of whether "now" can have a universal meaning--and the even subtler ontological question of what it means to exist--it makes sense to think of the totality of space and all of its contents at the present time, and to imagine this totality as a contiguous entity. If we take this route, we may first notice that space appears to us to be three-dimensional. Thus, we could make the assumption that we can locate anything in the universe using three Cartesian coordinates: at this frozen moment in time that we call the present, every object occupies a certain x , y and z in our three-dimensional continuum.

So here is one natural notion of the universe: all of three-dimensional space at the present time. Call it the nowverse. Fanciful theoretical constructs such as string theory postulate that, in fact, there is more to space than we can see, but for now those theories have no experimental evidence to support them. So, for the time being we may as well just focus on our familiar three dimensions. Time, on the other hand, is indeed an additional dimension, and together with space it forms a larger, four-dimensional entity called spacetime.


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It is natural to think of the nowverse as a 3-D slice in this 4-D space, just like horizontal planes are 2-D slices in our 3-D world. Because most people including yours truly have a hard time visualizing 4-D objects, a common way of thinking of spacetime is to pretend that space had only two dimensions.

Spacetime, then, would have a more manageable total of three. In this way of looking at things, the nowverse is one of many parallel planes, each of which represent the universe at a particular time of its history. For all we know, space is 3-D, and spacetime is 4-D; but if string theory is true, then space turns out to be 9-D, and spacetime D. Incidentally, when cosmologists talk about the expansion of the universe, they mean that space has been expanding, not spacetime.

In the last decade—you may have read this news countless times—cosmologists have found what they say is rather convincing evidence that the universe meaning 3-D space is flat, or at least very close to being flat. For the time being, it is convenient to just visualize a plane as our archetype of flat object, and the surface of the Earth as our archetype of a curved one. Both are two-dimensional, but as I will describe in the next installment, flatness and curviness make sense in any number of dimensions.

When cosmologists say that the universe is flat they are referring to space—the nowverse and its parallel siblings of time past. Spacetime is not flat. To put it succintly: space can be flat even if spacetime isn't. Moreover, when they talk about the flatness of space cosmologists are referring to the large-scale appearance of the universe.

Remarkable fresh evidence for this fact was obtained recently by the longest-running experiment in NASA history, Gravity Probe B , which took a direct measurement of the curvature of space around Earth. On a cosmic scale, the curvature created in space by the countless stars, black holes, dust clouds, galaxies, and so on constitutes just a bunch of little bumps on a space that is, overall, boringly flat.

If everything in the nowverse has an x , a y and a z, it would be natural to assume that we can push these coordinates to take any value, no matter how large. After all, what could stop her? The fact that you can go on forever however does not mean that space is infinite. Think of the two-dimensional sphere on which we live, the surface of the Earth. Something similar could, in principle, happen in our universe: a spaceship that flew off in one direction could, after a long time, reappear from the opposite direction.

Cosmologists seem to believe that the universe goes on forever without coming back—and in particular, that space has infinite extension. But when pressed, most cosmologists would also admit that, in fact, they have no clue whether it's finite or infinite. In principle, the universe could be finite and without a boundary—just like the surface of the Earth, but in three dimensions. In fact, when Einstein formulated his cosmological vision, based on his theory of gravitation, he postulated that the universe was finite.

Some have suggested that the way Dante describes the universe in his Divine Comedy has something to do with a 3-D sphere, too: I guess that will have to wait for a future post, too. Weeks; Scientific American , April ]. In a universe that has one of these shapes, one could observe trippy hall-0f-mirror type of effects.

The universe is For reasons that would be too complicated to go into here, that maximum distance is actually not So one thing we know is what we cannot know: the universe we can observe has finite extension. Cosmologists often refer to it as the observable universe. How large is the observable universe? That is a surprisingly difficult question, which will be the subject of yet another future post. Because the universe meaning space has been expanding ever since, those galaxies are now at a much greater distance—some 26 billion light-years away.

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Even farther away than the farthest galaxies, the most distant object we have been able to observe, the plasma that existed before the age of recombination [see Under a Blood Red Sky ], existed about The change in the shape of the stability potential, in this case, is due to the behavior of the meson mass in the minima of As Eq.

In this work we studied a sine-Gordon-like model, which is controlled by a real parameter that continuously connects the sine-Gordon and the vacuumless models. In the 5-dimensional case, we considered a warped geometry with a single extra dimension of infinite extent and studied the new braneworld scenario described in the small sector. Skip to main content Skip to sections. Advertisement Hide. Download PDF. From sine-Gordon to vacuumless systems in flat and curved spacetimes. Authors Authors and affiliations D. Bazeia D. Open Access. First Online: 18 December Relevant phenomena occur when the system under analysis presents a set of degenerate minima.

In these situations each pair of consecutive minima form distinct topological sectors that, in turn, have different solutions. These solutions are called kinks. An interesting way to characterize a kink is the existence of a topological current. Once we know the behavior of a given model, with its characteristics and general behavior, it is interesting to look for new well-behaved models. In this sense, the deformation method [ 12 ] is a powerful tool to find new models in field theory. Open image in new window. The second set of solutions we find from Eq.

We can now perform the analysis of the energy densities of the model. In this section we analyze the stability of the solutions of the models presented so far. The translational invariance of the solutions we presented so far implies the existence of at least one bound state for each topological sector, which is given by the zero mode of Eq.

Models described by scalar fields have direct applications in gravitation, providing braneworld scenarios for thick branes. In this context, the scalar field acts as a source of gravity around the brane, and thus describes how gravity behaves throughout the bulk. The analysis of the thick branes scenario generated by the large kink 27 is similar to the case of small kinks 29 , so we concentrate on the brane generated by the small kink. With these ingredients we analytically solve Eq. From Eq.

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In this section we analyze the stability of the gravitational sector. Vilenkin, E. Manton, P. Caudrey, J. Eilbeck, J.

general relativity

Cho, A. Vilenkin, Phys. Bazeia, Phys. Bazeia, F. Brito, J. Nascimento, Phys. Amaro de Faria Jr. D 72, Google Scholar. Brito, L. Brito, F. Costa, Phys. Bazeia, L.

Curved space - Wikipedia

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Bazeia, D. Moreira, Phys. Bazeia, M. Marques, R. Menezes, D. Moreira, Ann. Menezes, Phys. Bazeia, R. Menezes, Roldao da Rocha, Adv. High Energy Phys. Brito, M.

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