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Kaluza-Klein Compactification of Closed Strings
-----------------------------------------------
For a closed string at H, the angular momentum in the
compactified dimensions, y, around the cylinder is:
L = p x r = nh where r is the radius of compactification.
Therefore, setting m = 1 we get:
p = nh/r
The energy is given by:
E = pc = mc^{2} = cnh/r
Therefore,
m^{2} = n^{2}h^{2}/cr^{2}
This referred to as a KALUZA-KLEIN particle.
For the closed string wound around the cylinder in the
compactified dimensions, the energy is:
E = mc^{2} = 2πwrT
Where T is the tension (the energy per unit length) and w is
the WINDING NUMBER equal to the number of times the string is
wound around the cylinder in the compactified dimensions.
Therefore,
m^{2} = (2πwrT)^{2}/c^{4}
^{ } = w^{2}r^{2}/α'^{2}c^{4}
Where α' = 1/2πT is the string length.
The total mass squared (setting h = c = 1) is given by:
m^{2} = n^{2}/r^{2} + w^{2}r^{2}/α'^{2} + energy (mass) from oscillator modes.
Ignoring the last term for now, it is possible to identify 2
different mass scales for a particle. A very crucial feature
of this is that the quantized momentum modes of a closed string
are not distinguishable from the the winding modes of the string
in the compact dimension. This creates a symmetry between small
and large distances called T-DUALITY.
T-duality is a particular example of the idea of DUALITY in
physics. Duality refers to a situation where two seemingly
different physical systems turn out to be equivalent to each
other. If two theories are related by a duality, it means
that one theory can be transformed in some way so that it ends
up looking just like the other theory. In other words, the two
theories are mathematically different descriptions of the same
phenomena.
Here we have one theory strings propagating in spacetime shaped
like a cylinder of radius, r, while in the other theory strings
are propagating on a spacetime shaped like a cylinder of radius
1/r. The two theories are equivalent in the sense that all
observable quantities in one description are identified with
quantities in the dual description. In this case, the momentum
in one description takes discrete values that are equal to the
number of times the string winds around the circle in the dual
description. Therefore, T-Duality for CLOSED strings can be
summarized as follows:
n <-> w and 1/r <-> r
General Relativity in 5 Dimensions
----------------------------------
Kaluza–Klein theory is a unified field theory of gravitation
and electromagnetism built around the idea of a 5th dimension
beyond the 4 of spacetime. The theory hypothesises a 5D metric
of the form:
_{ } - -
g'_{ab} ≡ | g_{μν} + φ^{2} φ^{2}A_{μ} |
_{ } | φ^{2}A_{ν} _{ }φ^{2} _{ }|
_{ } - -
_{ } - -
_{ } ≡ | g'_{μν} g'_{5ν} |
_{ } | g'_{ν5} g'_{55} |
_{ } - -
Where A_{μ} is the vector potential and φ is the scalar potential.
a and b span 5 dimensions and so we get a 4D g_{μν} with an extra
dimension defined in terms of A_{μ} and φ. We can write:
g'_{μν} = g_{μν} + φ^{2}, g'_{5ν} = g_{ν5} = φ^{2}A_{μ} and g'_{55} = φ^{2}
The associated inverse metric is,
_{ } - -
g'^{ab} ≡ | g^{μν} -A^{μ}^{ } |
^{ } | -A^{ν} g_{αβ}A^{α}A^{β} + 1/φ^{2} |
^{ } - -
This metric implies:
ds^{2} = g'_{ab}dx^{a}dx^{b} = g_{μν}dx^{μ}dx^{ν} + φ^{2}(A_{ν}dx^{ν} + dx^{5})^{2}
The Einstein field equations are obtained by using the 5D
metric to construct 5D Christoffel symbols and then using
these to construct the 5D Ricci tensor. The solutions to
the field equations are most easily found for the the vacuum
case i.e. where the presence of matter is not considered and
the space is Ricci flat, meaning R'_{ab} = 0. Also, to simplify
the analysis, Kaluza also introduced the hypothesis known as
the "cylinder condition", ∂g'_{ab}/∂x^{5} = 0, that no component of
the 5D metric depends on the fifth dimension. Without this
assumption, the field equations of 5D relativity are enormously
more complex. Later, Klein explained the cylinder condition by
hypothesizing that the 5th dimension was both 'curled up' and
microscopic. Kaluza also set φ equal to a constant. Under
these conditions the solutions can be shown to be:
R'_{55} = 0 leads to:
g^{μν}∇_{μ}∇_{ν}φ = (1/4)φ^{3}F^{αβ}F_{αβ}
It shows that the electromagnetic field is a source for φ.
R'_{5α} = 0 leads to:
(1/2)g^{βμ}∇_{μ}(φ^{3}F_{αβ}) = 0
It has the form of the vacuum Maxwell equations if the scalar
field is constant.
R'_{μν} - (1/2)g'_{μν}R' = 0 leads to:
R_{μν} - (1/2)g_{μν}R = (1/2)φ^{2}(g^{αβ}F_{μα}F_{νβ} - (1/4)g_{μν}F_{αβ}F^{αβ})
+ (1/φ)(∇_{μ}∇_{ν}φ - g_{μν}g^{μν}∇_{μ}∇_{ν}φ)
Where ∇_{μ} is the covariant derivative from GR and F_{μα} is the
electromagnetic tensor.
This last equation shows that the electromagnetic tensor that
emerges from the 5D vacuum solutions is a source for the 4D
equations of gravity. This is a remarkable result. The right
hand side of Einstein's field equations in 4D is the stress-
energy-momentum tensor, so it seems there is an equivalence
between the electromagnetic field (charge) and the flow of
momentum of the Kaluza-Klein particle (H in the above diagram)
in the 5th dimension. In other words, the electromagnetic field
with charge as the source corresponds to the gravitation forces
of the 5 dimensional theory.
But what about the winding number? Is there a corresponding
field to the electromagnetic field whose source are winding
numbers instead of momenta. It turns out there is. The field
is called the KALB-RAMOND field, B_{μν}.
The Kalb-Ramond field is another form of the vector potential,
A^{μ} that has two indices instead of one. This difference is
related to the fact that while A^{μ} is integrated over the world
lines of particles to get its contributions to the action, the
Kalb–Ramond field must be integrated over the 2D worldsheet of
the string. Thus, while the action for a charged particle moving
in A_{μ} has the form:
S = -q∫dx^{μ}A_{μ}
A string coupled to the Kalb–Ramond field has the form:
S = -∫dx^{μ}dx^{ν}B_{μν}
So we can summarize by saying that T duality also exists for
charge. On the one hand there is an electromagnetic field
associated whose sources are the momenta. On the other hand
there is an analagous field, the Kalb-Ramond field, whose
sources are the winding numbers. In both cases, the fields
correspond to the gravitation forces of the 5th dimensional
theory.
The attraction and repulsion between strings originates
with the sign of the momentum quantum number, n, and the
sign of the winding number, w. Strings with opposite
momenta and winding number will attract each other while
strings with the same signs will repel each other.