User:TimothyRias/sandbox

Alternative coordinates
The Schwarzschild solution can be expressed in a range of different choices of coordinates besides the Schwarzschild coordinates used above. Different choices tend the highlight different features of the solution. The table below shows some popular choices.

In table above, some shorthand has been introduced for brevity. The speed of light c has been set to one. The notion $$ d\Omega^2= d\theta^2+\sin(\theta)^2 d\phi^2$$ is used for the metric of a two dimensional sphere. Moreover, in each entry R and T denote alternative choices of radial and time coordinate for the particular coordinates. Note, the R and/or T may vary from entry to entry.

Scientific impact
The existence of the Higgs field is the last prediction of the Standard Model that has not been tested experimentally. Finding the Higgs boson would at last prove the existence of this corner stone of one of the most successful theories of modern physics. Furthermore, if the Higgs boson is detected, its precise properties may allow us to distinguish between various possible extensions of the Standard Model such as supersymmetry.

The Higgs field plays various important roles in particle physics and possibly extending to cosmology.

Electroweak symmetry breaking
Historically the main reason to introduce the Higgs field has been to explain why the symmetry rules that govern the weak force are not precisely obeyed by nature, with as a main consequence—as explained by the Higgs mechanism—that the particles transmitting the weak force (W and Z bosons) are massive and that the weak force only acts at very short ranges.

If the Higgs field did not exist, the world would be a very different place. Weak symmetry would still be broken, but the W and Z bosons would be much lighter and their would be no pions. 

Mass of the elementary constituents of matter
In addition to explaining why the W and Z bosons are massive, the existence of the Higgs field also explains why the fundamental constituents of matter—such as quarks and electrons—have mass. Without the Higgs field, the Standard Model predicts that the electrons should be massless, and as a consequence it would be impossible for electron to be bound to a nucleus to form an atom, leading to a dramatically different universe.

It is important to note that the Higgs field is not responsible for all mass in the universe. For example, only about 1% of the mass of baryons (composite particles such as the proton and neutron) is due to the Higgs mechanism acting to produce the mass of quarks. The rest is due to the mass added by the kinetic energies of quarks and the energies of (massless) gluons of the strong interaction inside the baryons. Nonetheless, this small amount still has a significant impact. If the quarks making up protons and neutrons were massless, the neutron would be lighter than the proton and the laws of quantum physics would predict that all protons decay to neutrons, before hydrogen atoms could be formed.

The Higgs field also does not generate mas out of nothing. Mass simply is property that results from the presence of energy in a system. The Higgs field when interacting with other particles simply bestows them with some of the energy stored in the ground state of the field which manifests as their mass.

Higgs inflation
In cosmology, the remarkable homogeneity of the universe is explained by positing that the Universe has undergone a phase of exponential expansion known as inflation in its earliest moments. Such a phase requires the existence of a scalar field with a non-zero energy density everywhere, known as an inflaton. Since the Standard Model already contains such a fundamental scalar field in the form of the Higgs field, it has long been suggested the Higgs could also function as an inflaton.

Now, this hypothesis is largely discarded because it is difficult to reconcile with the precision measurements of the cosmic microwave background done by WMAP. Nonetheless, it has been argued that this difficulties can be avoided if the Higgs field has a strong non-minimal coupling to gravity.

Cosmological constant problem



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 * Validating the Standard Model, or choosing between extensions and alternatives
 * Finding how symmetry breaking happens within the electroweak interaction
 * Below an extremely high temperature, the electroweak interaction divides into electromagnetism and the weak force (known as "electroweak symmetry breaking"). Without this, the universe we see around us could not exist, because atoms and other structures could not form, and reactions in stars such as our Sun would not occur. But it is not confirmed how this actually happens. Is the Standard Model correct in its approach, and can it be made more exact with actual experimental measurements? If not the Higgs field, then what is breaking symmetry in its place?
 * Finding how certain particles acquire mass
 * The Higgs field and boson are often described as "creating mass", but this is only partially accurate. The Higgs field is the cause of mass for some–not all–particles, and it does not create mass out of nothing (which would violate the law of conservation of energy).
 * Below an extremely high temperature, the electroweak interaction divides into electromagnetism and the weak force (known as "electroweak symmetry breaking"). Without this, the universe we see around us could not exist, because atoms and other structures could not form, and reactions in stars such as our Sun would not occur. But it is not confirmed how this actually happens. Is the Standard Model correct in its approach, and can it be made more exact with actual experimental measurements? If not the Higgs field, then what is breaking symmetry in its place?
 * Finding how certain particles acquire mass
 * The Higgs field and boson are often described as "creating mass", but this is only partially accurate. The Higgs field is the cause of mass for some–not all–particles, and it does not create mass out of nothing (which would violate the law of conservation of energy).
 * The Higgs field and boson are often described as "creating mass", but this is only partially accurate. The Higgs field is the cause of mass for some–not all–particles, and it does not create mass out of nothing (which would violate the law of conservation of energy).

Electroweak symmetry breaking (due to a Higgs field or otherwise) is believed proven and responsible for the masses of particles such as fermions and the massive W and Z gauge bosons, and for their transition from massless to massive below a certain extremely high temperature (believed around 1015 K). But it is not responsible for the masses of many other particles. For example, only about 1% of the mass of baryons (composite particles such as the proton and neutron) is due to the Higgs mechanism acting to produce the mass of quarks. The rest is due to the mass added by the kinetic energies of quarks and the energies of (massless) gluons of the strong interaction inside the baryons.

The Standard Model shows how the energy of the Higgs field and vacuum can manifest, in the right conditions, as the property we call 'mass'. But as physicist and writer Max Jammer observes, in Higgs-based theories the Higgs field is not actually "creating" mass, and mass is not "generated" miraculously out of nothing. More accurately, mass is a manifestation of potential energy transferred to the particle during interactions ("coupling") with the Higgs field, which had contained that mass in the form of energy.
 * Evidence whether or not scalar fields exist in nature, and "new" physics
 * Proof of a scalar field such as the Higgs field would be hard to over estimate: "[The] verification of real scalar fields would be nearly as important as its role in generating mass". Rolf-Dieter Heuer, director general of the LHC project, stated in a 2011 talk on the Higgs field:
 * "All the matter particles are spin-1/2 fermions. All the force carriers are spin-1 bosons. Higgs particles are spin-0 bosons (scalars). The Higgs is neither matter nor force. The Higgs is just different. This would be the first fundamental scalar ever discovered. The Higgs field is thought to fill the entire universe. Could it give some handle of dark energy (scalar field)? Many modern theories predict other scalar particles like the Higgs. Why, after all, should the Higgs be the only one of its kind? [The] LHC can search for and study new scalars with precision."
 * "All the matter particles are spin-1/2 fermions. All the force carriers are spin-1 bosons. Higgs particles are spin-0 bosons (scalars). The Higgs is neither matter nor force. The Higgs is just different. This would be the first fundamental scalar ever discovered. The Higgs field is thought to fill the entire universe. Could it give some handle of dark energy (scalar field)? Many modern theories predict other scalar particles like the Higgs. Why, after all, should the Higgs be the only one of its kind? [The] LHC can search for and study new scalars with precision."


 * Insight into cosmic inflation
 * There has been considerable scientific research on possible links between the Higgs field and the inflaton - a hypothetical field suggested as the explanation for the expansion of space during the first fraction of a second of the universe (known as the "inflationary epoch"). Some theories suggest that a fundamental scalar field might be responsible for this phenomenon; the Higgs field is such a field and therefore has led to papers analysing whether it could also be the inflaton responsible for this exponential expansion of the universe during the Big Bang. Such theories are highly tentative and face significant problems related to unitarity, but may be viable if combined with additional features such as large non-minimal coupling, a Brans-Dicke scalar, or other "new" physics, and have received treatments suggesting that Higgs inflation models are still of interest theoretically.
 * Insight into the 'energy of the vacuum'
 * More speculatively, the Higgs field has also been proposed as the energy of the vacuum, which at the extreme energies of the first moments of the Big Bang caused the universe to be a kind of featureless symmetry of undifferentiated extremely high energy. In this kind of speculation, the single unified field of a Grand Unified Theory is identified as (or modeled upon) the Higgs field, and it is through successive symmetry breakings of the Higgs field or some similar field at phase transitions that the present universe's known forces and fields arise.
 * Link to the 'cosmological constant' problem
 * The relationship (if any) between the Higgs field and the presently observed vacuum energy density of the universe has also come under scientific study. As observed, the present vacuum energy density is extremely close to zero, but the energy density expected from the Higgs field, supersymmetry, and other current theories are typically many orders of magnitude larger. It is unclear how these should be reconciled. This cosmological constant problem remains a further major unanswered problem in physics.
 * Link to the 'cosmological constant' problem
 * The relationship (if any) between the Higgs field and the presently observed vacuum energy density of the universe has also come under scientific study. As observed, the present vacuum energy density is extremely close to zero, but the energy density expected from the Higgs field, supersymmetry, and other current theories are typically many orders of magnitude larger. It is unclear how these should be reconciled. This cosmological constant problem remains a further major unanswered problem in physics.
 * The relationship (if any) between the Higgs field and the presently observed vacuum energy density of the universe has also come under scientific study. As observed, the present vacuum energy density is extremely close to zero, but the energy density expected from the Higgs field, supersymmetry, and other current theories are typically many orders of magnitude larger. It is unclear how these should be reconciled. This cosmological constant problem remains a further major unanswered problem in physics.


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