Dr. N.L. Kachhara
Think of a very large ball (like Earth). Even though you look at the ball in three space dimensions, the outer surface of the ball has the geometry of a sphere in two dimensions, because there are only two independent directions on motion along the surface. If you were very small and lived on the surface of the ball, you might think you weren't on a ball at all, but on a big flat two- dimensional plane. But if you were to carefully measure distances on the sphere, you would discover that you were not living on a flat surface but on the curved surface of a large sphere.
The idea of the curvature of the surface of the ball can apply to the whole universe at once. That was the great breakthrough in Einstein's theory of general relativity. Space and time are unified into a single geometric entity called space-time, and the space-time has geometry, spacetime can be curved just like the surface of a large ball is curved.
The Einstein equation is the classical equation of motion for space-time, because quantum behaviour is never considered. For this reason, it is at best an approximation to exact theory. The Einstein Equation says that the curvature in space-time in a given direction is directly related to the energy and momentum of everything in the space-time that isn't space-time itself. In other words, the Einstein equation is what ties gravity to non-gravity, geometry to non-geometry. The curvature is the gravity, and all of the "other stuff" the electrons and quarks that make up the atoms that make up matter, the electromagnetic radiation, and every particle that mediates every force that isn't gravity lives in the curved space-time and at the same time determines its curvature through the Einstein equation.
The full description of a given space-time includes not only all space but also all of time; in other words, everything that happened and will ever happen in that space-time. To deal with the problem scientists make approximations and abstraction; they make abstract models that approximate the real universe fairly well at large distance, say at the scale of galactic clusters.
To solve the equations, simplifying assumptions also have to be made about the space-time curvature. The first assumption made is that space-time can be neatly separated into space and time. This assumption is well justified. The next important assumption, the one behind the Big Bang theory, is that at every time in universe, space looks the same in every direction at every point. So they are assuming that space is homogenous and isotopic. Cosmologists call this the assumption of maximal symmetry. At the large distance scales relevant to cosmology, it turns out that it is a reasonable approximation to make.
When cosmologists solve the Einstein equation for the space-time geometry of our universe, they consider three basic types of energy that could curve space-time: vacuum energy, radiation and matter. The radiation and matter in the universe are treated like uniform gases with equation of state that relate pressure to density. Once the assumptions of uniform energy sources and maximal symmetry of space have been made, the Einstein equation reduces to two ordinary differential equations that are easy to solve using basic calculus. The solutions tell us two things: the geometry of space, and how the size of space changes with time.
If at every time, space at every point looks the same in every direction; the space has to have constant curvature. If the curvature were different at any point, then space would look different in that direction from every other point. Therefore if space is maximally symmetric, the curvature has to be the same at every point. So that narrows us down to three options for the geometry of space: positive, negative or zero curvature when there is no vacuum energy present, just matter or radiation, the curvature of space also tells us the time evolution of the space-time in question.
- Positive curvature: The unique N-dimensional space with constant positive curvature is an N-dimensional sphere, a closed universe as shown in fig. 6.11 In this space-time, space expands from zero volume in a Big Bang but then reaches a maximum volume and starts to contract back to zero volume in a Big Crunch.
- Zero curvature: A space with zero curvature is called a flat surface. A flat space is noncompact, space extends infinitely far in any direction, so this option also represents an open universe. This space-time has space expanding forever in time.
- Negative curvature: The unique N-dimensional space with constant negative curvature is an N-dimensional pseudo sphere, a hyperboloid as shown in fig 6.12. With negative curvature, space has infinite volume and an open universe. This space-time also has space expanding forever in time.
What determines whether a Universe is open or closed? For a closed universe, the total energy density in the Universe has to be greater than the value that gives a flat universe, called critical density. So a closed Universe has Ω>1, a flat universe has Ω=1 and an open Universe has Ω<1.
The above analysis only takes into account energy and matter and neglects any vacuum energy that might be present. Vacuum energy leads to a constant energy density that is called the cosmological constant Λ.
Einstein did not always like the conclusions of his own work, His equation of motion of space-time predicted that a universe filled with ordinary matter would expand. To fix the Einstein equation, he added a term now called the cosmological constant that balanced the energy density of matter and radiation to make a universe that neither expanded nor contracted, but stayed the same for eternity. A cosmological constant can act to speed up or slow down the expansion of the universe, depending on whether it is positive or negative. When a cosmology constant is added to a space-time with matter and radiation, the story gets more complicated than the simple open or closed scenarios described above. Although many scientists including Einstein, had speculated that Λ was zero, recent astronomical observations have detected a large amount of dark energy that is accelerating the universe's expansion preliminary studies suggest that this dark energy corresponds to a positive.
The ultimate fate of the universe is still unknown, since it depends critically on the curvature and the cosmological constant. If the universe is sufficiently dense, its average curvature throughout is positive and the universe will eventually re collapse in a Big Crunch, possibly starting a new universe in a Big Bounce. Conversely, if the universe is insufficiently dense, the universe will expand for ever, cooling off and eventually becoming inhospitable for all life, as the stars die and all matter coalesces into black holes. Recent data suggests that the expansion of the universe is not decreasing as originally expected, but accelerating; if this continues indefinitely, the universe will eventually rip itself to shreds (the Big Rip). Experimentally, the universe has an overall density that is very close to the critical value between re collapse and eternal expansion; more careful astronomical observations are needed to decide the question. A recent measurement shows that the universe is flat with only a 2% margin of error.
