The cosmological constant is not the only way to get an accelerated expansion of the universe in alternatives to GR . We've already seen how the scalar potential \ lambda ( \ phi ) \; can be added to scalar tensor theories.
42.
In 3 dimensions, an exact vector field ( thought of as a 1-form ) is called a conservative vector field, meaning that it is the derivative ( gradient ) of a 0-form ( function ), called the scalar potential.
43.
For one charged particle of electric charge in an electromagnetic field, given by the magnetic vector potential defined by the magnetic field & nabla; & times; "'A "'} }, and electric scalar potential, this is:
44.
We can, if we desire, derive the accelerations a b and the adjusted " acceleration at infinity " ainf b from a scalar potential Z, though there is not necessarily any particular advantage in doing so . ( Wald 1984, pg 158, problem 4)
45.
Where } } is the vector potential for the magnetic field, } }; is the speed of light in vacuum; and ) } } is an arbitrary external scalar potential; one could easily take it to be quadratic in and, without loss of generality.
46.
The natural scale is na�vely the Planck scale so this small value could be seen as another form of fine-tuning ( called a hierarchy problem ) : the energy density given by the scalar potential is down by 10 & minus; 12 compared to the Planck density.
47.
"' Scalar potential "', simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other.
48.
:: : The reason a magnetic field's energy must be described with a vector potential, as opposed to a scalar potential, is because the force imparted by a magnetic field does not only depend on position ( it also depends on relative velocities and orientations ).
49.
In a 1990 paper, from the standpoint that the charged leptons are elementary, by introducing a scalar boson with ( octet + singlet ) of a family symmetry U ( 3 ), Koide re-derived the charged lepton mass formula from minimizing conditions for the scalar potential.
50.
In magnetostatics where there is no time-varying charge distribution, only the first equation is needed . ( In the context of electrodynamics, the terms " vector potential " and " scalar potential " are used for " magnetic vector potential " and " electric potential ", respectively.
