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In [[general relativity]], a '''vacuum solution''' is a [[Lorentzian manifold]] whose [[Einstein tensor]] vanishes identically.  According to the [[Einstein field equation]], this means that the [[stress-energy tensor]] also vanishes identically, so that no matter or non-gravitational fields are present.
 
More generally, a '''vacuum region''' in a Lorentzian manifold is a region in which the Einstein tensor vanishes.
 
==Equivalent conditions==
It is a mathematical fact that the Einstein tensor vanishes if and only if the [[Ricci tensor]] vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the '''trace reverse''' of each other:
:<math>G_{ab} = R_{ab} - \frac{R}{2} \, g_{ab}, \; \; R_{ab} = G_{ab} - \frac{G}{2} \, g_{ab}</math>
where the [[Trace (linear algebra)|trace]]s are <math>R = {R^a}_a, \; \; G = {G^a}_a = -R</math>.
 
A third equivalent condition follows from the [[Ricci decomposition]] of the [[Riemann curvature tensor]] as a sum of the [[Weyl curvature|Weyl curvature tensor]] plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree, <math>R_{abcd}=C_{abcd}</math>, in some region if and only if it is a vacuum region.
 
==Gravitational energy==
Since <math>T^{ab} = 0</math> in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no [[energy]].  But the gravitational field can do [[work (physics)|work]], so we must expect the gravitational field itself to possess energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest".
 
The fact that the gravitational field itself possesses energy yields a way to understand the nonlinearity of the Einstein field equation: this gravitational field energy itself produces more gravity.  This means that the gravitational field outside the Sun is a bit ''stronger'' according to general relativity than it is according to Newton's theory.
 
==Examples==
Well known examples of explicit vacuum solutions include:
 
*[[Minkowski spacetime]] (which describes empty space with no [[cosmological constant]])
*[[Milne model]] (which is a model developed by E. A. Milne describing an empty universe which has no curvature)
*[[Schwarzschild vacuum]] (which describes the spacetime geometry around a spherical mass),
*[[Kerr vacuum]] (which describes the geometry around a rotating object),
*[[Taub-NUT vacuum]] (a famous counterexample describing the exterior gravitational field of an isolated object with strange properties),
*[[Kerns–Wild vacuum]] (Robert M. Kerns and Walter J. Wild 1982) (a Schwarzschild object immersed in an ambient "almost uniform" gravitational field),
*[[double Kerr vacuum]] (two Kerr objects sharing the same axis of rotation, but held apart by unphysical zero active mass "cables" going out to suspension points infinitely removed),
*[[Khan–Penrose vacuum]] (K. A. Khan and [[Roger Penrose]] 1971) (a simple [[colliding plane wave]] model),
*[[Ozváth–Schücking metric|Oszváth–Schücking vacuum]] (the circularly polarized sinusoidal gravitational wave, another famous counterexample).
*[[Kasner metric]]
 
These all belong to one or more general families of solutions:
 
*the [[Weyl vacua]] ([[Hermann Weyl]]) (the family of all static vacuum solutions),
*the [[Beck vacua]] (Guido Beck 1925) (the family of all cylindrically symmetric nonrotating vacuum solutions),
*the [[Ernst vacua]] (Frederick J. Ernst 1968) (the family of all stationary axisymmetric vacuum solutions),
*the [[Ehlers vacua]] ([[Jürgen Ehlers]]) (the family of all cylindrically symmetric vacuum solutions),
*the [[Szekeres vacua]] ([[George Szekeres]]) (the family of all colliding gravitational plane wave models),
*the [[Gowdy vacua]] (Robert H. Gowdy) (cosmological models constructed using gravitational waves),
 
Several of the families mentioned here, members of which are obtained by solving an appropriate linear or nonlinear, real or complex partial differential equation, turn out to be very closely related, in perhaps surprising ways.
 
In addition to these, we also have the vacuum [[pp-wave spacetimes]], which include the [[gravitational plane wave]]s.
 
==See also==
* [[vacuum solution]], an article about vacuum solutions in physics in general,
* [[lambdavacuum solution]], an article about a significant generalization of the notion of a vacuum solution in general relativity,
* [[exact solutions in general relativity]], an article about all kinds of exact solutions to the Einstein field equation.
 
{{DEFAULTSORT:Vacuum Solution (General Relativity)}}
[[Category:Exact solutions in general relativity]]

Revision as of 04:40, 23 February 2014

Prostatic congestion and inflammation can stimulate the prostate urethra inducing frequent spermatorrhea. Despite this amzing incidence, few cases come to light. Coffee drank excessively causes a debilitating effect upon the sexual organs. In many cases, erectile dysfunction is curable and indeed in recent years the markets have been flooded with pills that can help men achieve and maintain erections strong. Ovulation means your ovaries are releasing an egg down your fallopian tubes and into your uterus, where the baby will stay for 9 months once it is conceived.

Here are some remedies for impotence you can try tonight. Tadalafil works by inhibiting the enzyme PDE-5, and helps to increase the blood supply to the penis. Most of the companies launch multiple products every year, and several new companies born every quarter. A few of the patterns truly flow straight down to the brim of the cap, although some have add-ons decorating the top like organizations or metallic studs, and sometimes several rhinestones. To stay healthy, it is critical to get peaceful rest for not less than 8 hrs on a daily basis.

Moreover, women are really keen on a slightly tipsy man, because usually it takes longer before he comes (if at all). It is ideal for food cooking, baking allowing it to do wonders to your own wellness, hair and skin. Several of the herbs taken with a healthy diet can help impotence. She recently wrote the best post on Tech - Crunch for months, a fascinating survey of Silicon Valley "mafias", and her latest book, Brilliant, Crazy, Cocky, is the result of forty weeks travelling around Asia, Africa and South America, conducting an exhaustive survey of entrepreneurialism in emerging economies. British researchers said pelvic floor exercises or Kegels, named after the South Californian gynecologist Dr.

This process involves an intraocular implant which is even popularly known as the iris implants and with the help of this you can easily change one's eye colour instead of using the color contacts or the lenses. It will help your stylist get an idea of what type of cut or color you may have in mind on a future visit. 4)One of the popular herbal remedies that you can rely on is the Gingko Biloba. PETA claims that meat eaters have higher rates of impotency due to artery blockage. Also, if you keep sitting in one position throughout the day, the chances are rife that you could suffer from impotency too.

This included eating healthy diet, daily exercise, and adequate sleep. Some of these habits include excessive drinking and smoking. Generic online pharmacy provides prescription drugs including ED Medicine which is now reaching distinctive horizons by effectively treating male impotence. This is why Erectile Dysfunction Drugs were created. It is also the official gemstone of the Zodiac sign Aries.

If you have any questions with regards to the place and how to use vitamins for erectile dysfunction (sneak a peek at these guys), you can speak to us at our web site.

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