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In [[special relativity]], '''four-momentum''' is the generalization of the classical three-dimensional [[momentum]] to four-dimensional [[spacetime]]. Momentum is a vector in three dimensions; similarly four-momentum is a [[four-vector]] in [[spacetime]]. The [[contravariant vector|contravariant]] four-momentum of a particle with three-momentum '''p''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'') and energy ''E'' is
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:<math>
\mathbf{P} = \begin{pmatrix}
P^0 \\ P^1 \\ P^2 \\ P^3
\end{pmatrix} =
\begin{pmatrix}
E/c \\ p_x \\ p_y \\ p_z
\end{pmatrix}
</math>
 
The four-momentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under [[Lorentz transformation]]s.
 
The above definition applies under the coordinate convention that ''x''<sup>0</sup> = ''ct''. Some authors use the convention ''x''<sup>0</sup> = ''t'' which yields a modified definition with ''P''<sup>0</sup> = ''E''/''c''<sup>2</sup>. It is also possible to define [[covariant vector|covariant]] four-momentum ''P''<sub>μ</sub> where the sign of the energy is reversed.
 
== Minkowski norm ==
 
Calculating the [[Minkowski space#Structure|Minkowski norm]] of the four-momentum gives a [[Lorentz invariant]] quantity equal (up to factors of the [[speed of light]] ''c'') to the square of the particle's [[proper mass]]:
:<math>-\|\mathbf{P}\|^2 = - P^\mu P_\mu = - \eta_{\mu\nu} P^\mu P^\nu = {E^2 \over c^2} - |\mathbf p|^2 = m^2c^2 </math>
 
where we use the convention that
:<math>\eta_{\mu\nu} = \begin{pmatrix}
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}</math>
 
is the metric tensor of [[special relativity]]. The magnitude ||'''P'''||<sup>2</sup> is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference.
 
== Relation to four-velocity ==
 
For a massive particle, the four-momentum is given by the particle's [[invariant mass]] ''m'' multiplied by the particle's [[four-velocity]]:
:<math>P^\mu = m \, U^\mu\!</math>
 
where the four-velocity is
:<math>
\begin{pmatrix}
U^0 \\ U^1 \\ U^2 \\ U^3
\end{pmatrix} =
\begin{pmatrix}
\gamma c \\ \gamma v_x \\ \gamma v_y \\ \gamma v_z
\end{pmatrix}
</math>
 
and
:<math>\gamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}</math>
is the [[Lorentz factor]], ''c'' is the [[speed of light]].
 
== Conservation of four-momentum ==
 
The conservation of the four-momentum yields two conservation laws for "classical" quantities:
# The total [[energy]] ''E'' = ''P''<sup>0</sup>''c'' is conserved.
# The classical three-[[momentum]] '''p''' is conserved.
 
Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since [[kinetic energy]] in the system center-of-mass frame and [[potential energy]] from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (−5 GeV/''c'', 4 GeV/''c'', 0, 0) and (−5 GeV/''c'', −4 GeV/''c'', 0, 0) each have (rest) mass 3 GeV/''c''<sup>2</sup> separately, but their total mass (the system mass) is 10 GeV/''c''<sup>2</sup>. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/''c''<sup>2</sup>.
 
One practical application from [[particle physics]] of the conservation of the [[invariant mass]] involves combining the four-momenta '''P'''(''A'') and '''P'''(''B'') of two daughter particles produced in the decay of a heavier particle with four-momentum '''P'''(''C'') to find the mass of the heavier particle. Conservation of four-momentum gives ''P''(''C'')<sup>μ</sup> = ''P''(''A'')<sup>μ</sup> + ''P''(''B'')<sup>μ</sup>, while the mass ''M'' of the heavier particle is given by −||'''P'''(''C'')||<sup>2</sup> = ''M''<sup>2</sup>''c''<sup>2</sup>. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for [[Z' boson]]s at high-energy particle [[collider]]s, where the Z' boson would show up as a bump in the invariant mass spectrum of [[electron]]-[[positron]] or [[muon]]-antimuon pairs.
 
If an object's mass does not change, the Minkowski inner product of its four-momentum and corresponding [[four-acceleration]] ''A''<sup>μ</sup> is zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so
:<math>P^{\mu} A_\mu = \eta_{\mu\nu} P^{\mu} A^\nu = \eta_{\mu\nu} P^\mu \frac{d}{d\tau} \frac{P^{\nu}}{m} = \frac{1}{2m} \frac{d}{d\tau} \|\mathbf{P}\|^2 = \frac{1}{2m} \frac{d}{d\tau} (-m^2c^2) = 0 .</math>
 
== Canonical momentum in the presence of an electromagnetic potential ==
For a [[charged particle]] of [[electric charge|charge]] ''q'', moving in an electromagnetic field given by the [[electromagnetic four-potential]]:
 
:<math>
\begin{pmatrix}
A^0 \\ A^1 \\ A^2 \\ A^3
\end{pmatrix} =
\begin{pmatrix}
\phi / c \\ A_x \\ A_y \\ A_z
\end{pmatrix}
</math>
 
where φ is the [[scalar potential]] and '''A''' = (''A<sub>x</sub>'', ''A<sub>y</sub>'', ''A<sub>z</sub>'') the [[vector potential]], the "canonical" momentum four-vector is
 
:<math> Q^\mu = P^\mu + q A^\mu. \!</math>
 
This allows the potential energy from the charged particle in an electrostatic potential and the [[Lorentz force]] on the charged particle moving in a magnetic field to be incorporated in a compact way, in [[relativistic quantum mechanics]].
 
==See also==
*[[Four-force]]
*[[Pauli–Lubanski pseudovector]]
 
== References ==
*{{cite book|last=Goldstein|first=Herbert|title=Classical mechanics|year=1980|publisher=Addison–Wesley Pub. Co.|location=Reading, Mass.|isbn=0201029189|edition=2nd |ref=harv}}
*{{cite book|last=Landau|first=L.D.|title=The classical theory of fields|year=2000|publisher=Butterworth Heinemann|location=Oxford|isbn=9780750627689 |coauthors=E.M. Lifshitz  |others=4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh|ref=harv}}
*{{cite book | author = Rindler, Wolfgang | title=Introduction to Special Relativity |edition=2nd| location= Oxford | publisher=Oxford University Press | year=1991 | isbn=0-19-853952-5 |ref=harv}}
 
[[Category:Minkowski spacetime]]
[[Category:Theory of relativity]]

Latest revision as of 03:54, 12 November 2014

A fast abdomen is the dream of all menopausal women. One of the many distressing effects of menopause is what appears to become the sudden appearance of unsightly and perhaps flabby stomach fat. Whenever you look in the mirror, you long for the days whenever a abdomen was flat plus fast. Not just does which belly weight change how you feel inside a clothing, it also puts we at risk for certain serious health challenges.

The final piece of the puzzle is to relax. Yes, I recognize you just said we should commit to working out because usually because possible, however, what we are speaking about here is steering away from stress because much as we potentially can. Stress seems to receive we achieving for snack foods for comfort, which just ends up adding to the weight problems you will absolutely be having. As we mentioned earlier, the perfect waist hip ratio is not an absolute necessity, but should you may get it as close because possible to that 0.70 number, you will virtually be inside amazing shape.

The 2nd measurement for we to take is your hip measuring. Well, let me clarify which. If your hips are wider than the waist, you'll need to take it. If a waist is greater, there's no need, and you waist hip ratio calculator can move on. If you do have to take the hip measuring, it's basically the same as the waist measurement, but it has to go around we at the widest point under your waist.

Why why a man swimmer's body is really sexy is because they have low body fat ratio so which their effectively toned muscles is enjoyed plus not covered by too much body fat. Although low in body fat, swimmers body fat ratio are not because low as professional bodybuilders throughout competition whose bodies have veins popping out with ripped railway track definitions. The swimmer's body is defined by Wisegeek as a significant amount of muscle tone with a lack of bulk plus is a universally attractive body form.

Burpees- This full body exercise is a advantageous method to indicate the fitness level in the event you may do even a few inside a row. A burpee is an intense exercise which challenges the muscular fitness, cardiovascular fitness, endurance plus body energy. Perform because many as possible inside one minute to certainly test and drive a fitness level.

Having which so called "apple" shape appears to bring higher risks than the "pear" shaped body, though nobody is certain why. It might be that fat stored in the abdomen sends out inflammatory substances, plus physicians recognize which inflammation is linked to heart difficulties. We do recognize a high waist hip ratio is linked to diabetes and hypertension, today it appears to be associated to sudden cardiac death too.

Other qualities and characteristics which females look for inside a man: Women wish dedication. We wish To recognize that a guy is interested inside their future. A man with a strong sense of personal financial, social and. Friendliness is seen as an almost irresistible train We hear women state, an interesting comment, because a guy that "wise with children" How significant is cash? Almost all the women interviewed inside the study following research shows that they wish a guy whom makes more funds. Women thrive and enjoy existence inside the experience of protection and stability. And last however, not least, a guy that cares for them plus their kids.What guys find irresistible in women?

What used to be called "middle-aged spread" has become a ubiquitous phenomenon across all age groups in America, including our children. It is time for us to take a closer look at the lifestyle, for it's shortening our lifetime spans.

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