Dear friends,

It is of course well-known that the (contravariant) spacetime metric
g^uv and Dirac gamma^u matrices are related by the commutation
relationship:

g^uv = (1/2) (gamma^u gamma^v + gamma^v gamma^u) (1)

where u = 0,1,2,3 are spacetime indices. When the gamma^u are precisely
equal to the well-known Dirac matrices which incorporate pairs of the
Pauli matrices in either the Dirac or Weyl representation in a
well-known manner, the g^uv arrived at via (1) are given by g^uv = n^uv
(the Minkowski metric), with diag (g_uv=n_uv) = (1,-1,-1,-1), and where
each g^uv=n^uv is actually a 4x4 diagonal unit matrix with diag(g_00)=1
and diag (g_kk)=-1 with Dirac spinor indexes implicitly suppressed.

Here are my questions:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv = n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?

2) Am I correct therefore, in thinking that in a gravitational field,
each of the Dirac matrices themselves ought to differ from
aforementioned the pairs of Pauli matrices as well, by some 4x4 spinor
matrix h^u? In other words, is to correct to think that in a
gravitational field, one may write:

gamma^u = n^u + K h^u, (2)

where n^u are the usual gamma^u utilizing Pauli matrix pairs and this
difference h^u is effectively another way of representing the
gravitational field h^uv within the gamma^u, with K being some constant
related to kappa?

3) Given equation (1), if the g^uv are in fact the g^uv = n^uv + kappa
h^uv which define the geometric curvature of spacetime, is it fair to
think of the gamma^u = n^u + K h^u as being just as capable of
representing spacetime curvature as g^uv = n^uv + kappa h^uv, and even
further, as capturing certain subtleties in spacetime structure because
of their spinor nature which the g^uv cannot alone capture? Put
differently, in general relativity we define the curvature of spacetime
by its metric. Can we equally think, and maybe even more fundamentally
think, that spacetime is alternatively defined by its gamma matrices
gamma^u, from which the g^uv may in turn be deduced by (1)? In other
words, can we think of the Dirac gammas as the "structure matrices of
spacetime" which, via (1), give us an alternative way to define a
classical spacetime metric?

4) The axial gamma^5 matrix is defined by gamma^5 = i gamma^0 gamma^1
gamma^2 gamma^3, and in this way, looks to be a "peer" of all the other
gamma^u. For example, we can equally write gamma^0 = i gamma^1 gamma^2
gamma^3 gamma^5, or gamma^1 = i gamma^0 gamma^2 gamma^3 gamma^5 or
gamma^3 = i gamma^0 gamma^1 gamma^2 gamma^5 or gamma^2 = -i gamma^0
gamma^1 gamma^3 gamma^5 (note that gamma^2 is the only matrix which
flips the sign when isolated). If the Dirac gamma^u can be thought of
as the "structure matrices of spacetime," and because gamma^5 appears to
be on a completely equal footing with the gamma^u, with u = 0,1,2,3, at
least via the relationship:

1 = i gamma^0 gamma^1 gamma^2 gamma^3 gamma^5 (3)

is there any reason why we cannot also think of the axial gamma^5 as a
fifth structure matrix of spacetime?

5) Is there any reason, therefore, why we ought not rewrite the
spacetime metric tensor (1), including the gamma^5, as:

g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (4)

with U = 0,1,2,3 and 5?

6) If we can fairly define a metric tensor g^UV with U = 0,1,2,3 and 5,
as in equation (4), does it make sense to conclude that the existence of
the gamma^5 is indicative of a fifth spacetime dimension?

7) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric g_UV =
n_UV defined by (4) then has:

diag (g_UV=n_UV) = (1,-1,-1,-1,1). (5)

This "fifth" dimension thereby naturally acquires a timelike signature.
Does it make any sense, therefore, to think of this fifth dimension
originating in gamma^5 as a second, "axial time" dimension?

8) If gamma^5 signifies a second time dimension, which yields a 5 = d =
D + 2 dimensional spacetime with D=3 space dimensions, then is there any
reason why we cannot define an invariant "mega-"proper time differential
interval dT in addition to the usual proper time invariant dtau, and use
g_UV to raise and lower indexes, according to:

dT^2 = g_UV dx^U dx^V =dx^U dx_U
= dx^u dx_u + dx^5 dx_5 = dtau^2 + dx^5 dx_5 ? (6)

9) Then just as we can perform ordinary rotations through the D=3 space
dimensions, is there anything which would bar us from considering a
rotation between the usual time dimension x^0 and an axial time
dimension x^5? Then, just as Feynman taught that particles, e.g.,
electrons can move "backwards" through time, might we also "define" the
reference frame of an observer as one in which dx_5=0, always, and
consider some particles as moving with a "sideways" component through
time, dx^5 not==0 relative an observer's movement through time? For
example, might we even think of a massless photon or graviton, for which
dtau=0, as moving fully sideways through time, with dT = dx^5 but
dtau=0?

10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?

Thanks,

Jay.
_____________________________
Jay R. Yablon
Email:

Jay R. Yablon wrote:
[...]
Here are my questions:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv = n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?

I'll be able to help you with this just as soon as another poster,
( somewhat less cryptic than yourself) shows us how to compute
the 'muzzle frequency' of a bullet. ;-)

http://nobelprize.org/physics/articl...ong/index.html
http://www.warwick.ac.uk/~phsbm/fqhe.htm

Sue...




2) Am I correct therefore, in thinking that in a gravitational field,
each of the Dirac matrices themselves ought to differ from
aforementioned the pairs of Pauli matrices as well, by some 4x4 spinor
matrix h^u? In other words, is to correct to think that in a
gravitational field, one may write:

gamma^u = n^u + K h^u, (2)

where n^u are the usual gamma^u utilizing Pauli matrix pairs and this
difference h^u is effectively another way of representing the
gravitational field h^uv within the gamma^u, with K being some constant
related to kappa?

3) Given equation (1), if the g^uv are in fact the g^uv = n^uv + kappa
h^uv which define the geometric curvature of spacetime, is it fair to
think of the gamma^u = n^u + K h^u as being just as capable of
representing spacetime curvature as g^uv = n^uv + kappa h^uv, and even
further, as capturing certain subtleties in spacetime structure because
of their spinor nature which the g^uv cannot alone capture? Put
differently, in general relativity we define the curvature of spacetime
by its metric. Can we equally think, and maybe even more fundamentally
think, that spacetime is alternatively defined by its gamma matrices
gamma^u, from which the g^uv may in turn be deduced by (1)? In other
words, can we think of the Dirac gammas as the "structure matrices of
spacetime" which, via (1), give us an alternative way to define a
classical spacetime metric?

4) The axial gamma^5 matrix is defined by gamma^5 = i gamma^0 gamma^1
gamma^2 gamma^3, and in this way, looks to be a "peer" of all the other
gamma^u. For example, we can equally write gamma^0 = i gamma^1 gamma^2
gamma^3 gamma^5, or gamma^1 = i gamma^0 gamma^2 gamma^3 gamma^5 or
gamma^3 = i gamma^0 gamma^1 gamma^2 gamma^5 or gamma^2 = -i gamma^0
gamma^1 gamma^3 gamma^5 (note that gamma^2 is the only matrix which
flips the sign when isolated). If the Dirac gamma^u can be thought of
as the "structure matrices of spacetime," and because gamma^5 appears to
be on a completely equal footing with the gamma^u, with u = 0,1,2,3, at
least via the relationship:

1 = i gamma^0 gamma^1 gamma^2 gamma^3 gamma^5 (3)

is there any reason why we cannot also think of the axial gamma^5 as a
fifth structure matrix of spacetime?

5) Is there any reason, therefore, why we ought not rewrite the
spacetime metric tensor (1), including the gamma^5, as:

g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (4)

with U = 0,1,2,3 and 5?

6) If we can fairly define a metric tensor g^UV with U = 0,1,2,3 and 5,
as in equation (4), does it make sense to conclude that the existence of
the gamma^5 is indicative of a fifth spacetime dimension?

7) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric g_UV =
n_UV defined by (4) then has:

diag (g_UV=n_UV) = (1,-1,-1,-1,1). (5)

This "fifth" dimension thereby naturally acquires a timelike signature.
Does it make any sense, therefore, to think of this fifth dimension
originating in gamma^5 as a second, "axial time" dimension?

8) If gamma^5 signifies a second time dimension, which yields a 5 = d =
D + 2 dimensional spacetime with D=3 space dimensions, then is there any
reason why we cannot define an invariant "mega-"proper time differential
interval dT in addition to the usual proper time invariant dtau, and use
g_UV to raise and lower indexes, according to:

dT^2 = g_UV dx^U dx^V =dx^U dx_U
= dx^u dx_u + dx^5 dx_5 = dtau^2 + dx^5 dx_5 ? (6)

9) Then just as we can perform ordinary rotations through the D=3 space
dimensions, is there anything which would bar us from considering a
rotation between the usual time dimension x^0 and an axial time
dimension x^5? Then, just as Feynman taught that particles, e.g.,
electrons can move "backwards" through time, might we also "define" the
reference frame of an observer as one in which dx_5=0, always, and
consider some particles as moving with a "sideways" component through
time, dx^5 not==0 relative an observer's movement through time? For
example, might we even think of a massless photon or graviton, for which
dtau=0, as moving fully sideways through time, with dT = dx^5 but
dtau=0?

10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?

Thanks,

Jay.
_____________________________
Jay R. Yablon
Email:

Jay R. Yablon wrote:
Dear friends,

It is of course well-known that the (contravariant) spacetime metric
g^uv and Dirac gamma^u matrices are related by the commutation
relationship:

g^uv = (1/2) (gamma^u gamma^v + gamma^v gamma^u) (1)

where u = 0,1,2,3 are spacetime indices. When the gamma^u are precisely
equal to the well-known Dirac matrices which incorporate pairs of the
Pauli matrices in either the Dirac or Weyl representation in a
well-known manner, the g^uv arrived at via (1) are given by g^uv = n^uv
(the Minkowski metric), with diag (g_uv=n_uv) = (1,-1,-1,-1), and where
each g^uv=n^uv is actually a 4x4 diagonal unit matrix with diag(g_00)=1
and diag (g_kk)=-1 with Dirac spinor indexes implicitly suppressed.

Here are my questions:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv = n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?

It can technically be any arbitrary metric. The literature is full of
papers on this.


2) Am I correct therefore, in thinking that in a gravitational field,
each of the Dirac matrices themselves ought to differ from
aforementioned the pairs of Pauli matrices as well, by some 4x4 spinor
matrix h^u? In other words, is to correct to think that in a
gravitational field, one may write:

gamma^u = n^u + K h^u, (2)

where n^u are the usual gamma^u utilizing Pauli matrix pairs and this
difference h^u is effectively another way of representing the
gravitational field h^uv within the gamma^u, with K being some constant
related to kappa?

Maybe not necessarily in that particular form, but you're on the right
track, as the gamma matrices will be different from the Lorentzian form
and have the same coordinate dependence as the metric. Again there is
plenty of literature relating to this.

3) Given equation (1), if the g^uv are in fact the g^uv = n^uv + kappa
h^uv which define the geometric curvature of spacetime, is it fair to
think of the gamma^u = n^u + K h^u as being just as capable of
representing spacetime curvature as g^uv = n^uv + kappa h^uv, and even
further, as capturing certain subtleties in spacetime structure because
of their spinor nature which the g^uv cannot alone capture? Put
differently, in general relativity we define the curvature of spacetime
by its metric. Can we equally think, and maybe even more fundamentally
think, that spacetime is alternatively defined by its gamma matrices
gamma^u, from which the g^uv may in turn be deduced by (1)? In other
words, can we think of the Dirac gammas as the "structure matrices of
spacetime" which, via (1), give us an alternative way to define a
classical spacetime metric?

Some people think that a spacetime based entirely on a modified gamma
in this regard would be a sufficient alternative to the traditional
metric-based geometry. You should be able to locate more info he

http://www.compukol.com/mendel/

4) The axial gamma^5 matrix is defined by gamma^5 = i gamma^0 gamma^1
gamma^2 gamma^3, and in this way, looks to be a "peer" of all the other
gamma^u. For example, we can equally write gamma^0 = i gamma^1 gamma^2
gamma^3 gamma^5, or gamma^1 = i gamma^0 gamma^2 gamma^3 gamma^5 or
gamma^3 = i gamma^0 gamma^1 gamma^2 gamma^5 or gamma^2 = -i gamma^0
gamma^1 gamma^3 gamma^5 (note that gamma^2 is the only matrix which
flips the sign when isolated). If the Dirac gamma^u can be thought of
as the "structure matrices of spacetime," and because gamma^5 appears to
be on a completely equal footing with the gamma^u, with u = 0,1,2,3, at
least via the relationship:

1 = i gamma^0 gamma^1 gamma^2 gamma^3 gamma^5 (3)

is there any reason why we cannot also think of the axial gamma^5 as a
fifth structure matrix of spacetime?

Gamma^5 is the unit pseudoscalar for four dimensional Clifford algebra.
In other words, it is just a number, but one that changes sign under
reflections. A four dimensional Clifford algebra possesses 16
indpendent elements: one scalar (1), 4 independent unit vectors
gamma^u, 6 independent unit bi-vectors 1/2 ( gamma^u gamma^v - gamma^v
gamma^v ), 4 independent pseudo-vectors gamma^5 gamma^u, and one unit
pseudoscalar gamma^5.

5) Is there any reason, therefore, why we ought not rewrite the
spacetime metric tensor (1), including the gamma^5, as:

g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (4)

with U = 0,1,2,3 and 5?

You might be able to get away with this, despite gamma^5 not
representing a vector. In fact, it might be interesting in that g^u5
is technically a pseudovector while g^55 is a scalar in the original
four dimensional world.

6) If we can fairly define a metric tensor g^UV with U = 0,1,2,3 and 5,
as in equation (4), does it make sense to conclude that the existence of
the gamma^5 is indicative of a fifth spacetime dimension?

Not necessarily, but I guess you'd be making it one.

7) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric g_UV =
n_UV defined by (4) then has:

diag (g_UV=n_UV) = (1,-1,-1,-1,1). (5)

This "fifth" dimension thereby naturally acquires a timelike signature.
Does it make any sense, therefore, to think of this fifth dimension
originating in gamma^5 as a second, "axial time" dimension?

It appears to have a timelike signature in 5-space, but it would have
been a scalar in 4-space, so it is not quite on the same footing.

8) If gamma^5 signifies a second time dimension, which yields a 5 = d =
D + 2 dimensional spacetime with D=3 space dimensions, then is there any
reason why we cannot define an invariant "mega-"proper time differential
interval dT in addition to the usual proper time invariant dtau, and use
g_UV to raise and lower indexes, according to:

dT^2 = g_UV dx^U dx^V =dx^U dx_U
= dx^u dx_u + dx^5 dx_5 = dtau^2 + dx^5 dx_5 ? (6)

I think you just did. No objection as far as I can see.

9) Then just as we can perform ordinary rotations through the D=3 space
dimensions, is there anything which would bar us from considering a
rotation between the usual time dimension x^0 and an axial time
dimension x^5? Then, just as Feynman taught that particles, e.g.,
electrons can move "backwards" through time, might we also "define" the
reference frame of an observer as one in which dx_5=0, always, and
consider some particles as moving with a "sideways" component through
time, dx^5 not==0 relative an observer's movement through time? For
example, might we even think of a massless photon or graviton, for which
dtau=0, as moving fully sideways through time, with dT = dx^5 but
dtau=0?

It's really hard to say how you could interpret this. But it could be
interesting to think about.

10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?

No objection to two timelike directions at least in the micro world.
In the classical world, it would have to disappear. In other words,
the contribution of gamma^5 to the metric would have to be
scale-dependent because it would have to die out at larger and larger
scales.

Hi Mr. Yablon,
I'm a bit rusty because I was outdoors this summer.
My replies will be "c" grade...

Jay R. Yablon wrote:
Dear friends,

It is of course well-known that the (contravariant) spacetime metric
g^uv and Dirac gamma^u matrices are related by the commutation
relationship:

g^uv = (1/2) (gamma^u gamma^v + gamma^v gamma^u) (1)

where u = 0,1,2,3 are spacetime indices. When the gamma^u are precisely
equal to the well-known Dirac matrices which incorporate pairs of the
Pauli matrices in either the Dirac or Weyl representation in a
well-known manner, the g^uv arrived at via (1) are given by g^uv = n^uv
(the Minkowski metric), with diag (g_uv=n_uv) = (1,-1,-1,-1),

That diag is a choice, I'm antsy about pre-selection,
because (1,1,1,1) can be used, that intro by AE back
in 1916 was a simplification as was sqr|g_uv| =-1.

and where
each g^uv=n^uv is actually a 4x4 diagonal unit matrix with diag(g_00)=1
and diag (g_kk)=-1 with Dirac spinor indexes implicitly suppressed.

Here are my questions:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv = n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?

IMO, the presences of matter, that reacts with the test
particle used to determine the "field" causes an apparent
"Non-Orthogonal" geometric relation on the trajectory of
the test particle, specifically, deviant from a cartesian
trajectory, and therefore the "weak field" correction "h^uv"
is applicable.
I'll add that the "h^uv" appears as a 1st order perturbation,
and in stronger fields, for example those encountered
within the nucleus itself, more accuruate terms would be
required, and more careful formula.

2) Am I correct therefore, in thinking that in a gravitational field,
each of the Dirac matrices themselves ought to differ from
aforementioned the pairs of Pauli matrices as well, by some 4x4 spinor
matrix h^u? In other words, is to correct to think that in a
gravitational field, one may write:

gamma^u = n^u + K h^u, (2)

IMO, yes, however we'll need two relating particles,
like "a" and "b" with a finite spacetime relation, ie,
the're at different places. Let me rewrite (2) as,

g(a)^u = n(a)^u + h(a)^u, (kst2)

g(b)^v = n(b)^v + h(b)^v,

then

g^uv = g(a)^u g(b)^v) (kst 3)

= n(a)^u n(b)^v + h(a)^u h(b)^v (kst 4)

The metric in Eq. (kst 3) is relational as it involves
the relation of "a" and "b".
I'm assuming we get Eq.(kst 4) by a "dot" product
using "h" perpendicular to "n".

where n^u are the usual gamma^u utilizing Pauli matrix pairs and this
difference h^u is effectively another way of representing the
gravitational field h^uv within the gamma^u, with K being some constant
related to kappa?

I think so, colloquially,

h^uv = h(a)^u h(b)^v

at either particle "a" or "b".

3) Given equation (1), if the g^uv are in fact the g^uv = n^uv + kappa
h^uv which define the geometric curvature of spacetime, is it fair to
think of the gamma^u = n^u + K h^u as being just as capable of
representing spacetime curvature as g^uv = n^uv + kappa h^uv, and even
further, as capturing certain subtleties in spacetime structure because
of their spinor nature which the g^uv cannot alone capture?

Once the metric is define relationally, which is reasonable
given our operational definition of time, (using a Cs clock
jumping a distance in time, (keep an eye on that reality)),
we're obliged to regard "relativity of measure" to require
a relation, so IMO, that requires Eq.(kst 4).

Put
differently, in general relativity we define the curvature of spacetime
by its metric. Can we equally think, and maybe even more fundamentally
think, that spacetime is alternatively defined by its gamma matrices
gamma^u, from which the g^uv may in turn be deduced by (1)? In other
words, can we think of the Dirac gammas as the "structure matrices of
spacetime" which, via (1), give us an alternative way to define a
classical spacetime metric?

The way I see it, is the Dirac gamma's are deduced
from a relation, specifically the 2nd rank tensor g_uv.
Interjecting, from the standpoint of QT, we'll need a
pair of charges to vary their energy and produce a
photon somewhere to provide a photograph.

4) The axial gamma^5 matrix is defined by gamma^5 = i gamma^0 gamma^1
gamma^2 gamma^3, and in this way, looks to be a "peer" of all the other
gamma^u. For example, we can equally write gamma^0 = i gamma^1 gamma^2
gamma^3 gamma^5, or gamma^1 = i gamma^0 gamma^2 gamma^3 gamma^5 or
gamma^3 = i gamma^0 gamma^1 gamma^2 gamma^5 or gamma^2 = -i gamma^0
gamma^1 gamma^3 gamma^5 (note that gamma^2 is the only matrix which
flips the sign when isolated). If the Dirac gamma^u can be thought of
as the "structure matrices of spacetime," and because gamma^5 appears to
be on a completely equal footing with the gamma^u, with u = 0,1,2,3, at
least via the relationship:

1 = i gamma^0 gamma^1 gamma^2 gamma^3 gamma^5 (3)

is there any reason why we cannot also think of the axial gamma^5 as a
fifth structure matrix of spacetime?

Well suppose working in 4D, I have a |g_uv| = g =/=1.
OK, that refracts light. The 5th D may provide a greater
elegance to the laws of physics.

5) Is there any reason, therefore, why we ought not rewrite the
spacetime metric tensor (1), including the gamma^5, as:

g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (4)

with U = 0,1,2,3 and 5?

LOL, maybe we should return to,
U= 1,2,3,4,5
and leave zero out of dimensionality!

6) If we can fairly define a metric tensor g^UV with U = 0,1,2,3 and 5,
as in equation (4), does it make sense to conclude that the existence of
the gamma^5 is indicative of a fifth spacetime dimension?

The two schools of thought I know are this,
1st: You may postulate a 5th.

2nd:I follow the nonsymetrical metric thou,
where g_uv =/= g_vu do to the apparent requirement
of the metric being defined relationationally, (as above).

So, my current research, in view of magnetism, is
to follow the 2nd.
I'll stop here for now, and I'll be taken my foot out of
my mouth before addressing the rest of your fine post.
Best Regards
Ken
...........

7) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric g_UV =
n_UV defined by (4) then has:

diag (g_UV=n_UV) = (1,-1,-1,-1,1). (5)

This "fifth" dimension thereby naturally acquires a timelike signature.
Does it make any sense, therefore, to think of this fifth dimension
originating in gamma^5 as a second, "axial time" dimension?

8) If gamma^5 signifies a second time dimension, which yields a 5 = d =
D + 2 dimensional spacetime with D=3 space dimensions, then is there any
reason why we cannot define an invariant "mega-"proper time differential
interval dT in addition to the usual proper time invariant dtau, and use
g_UV to raise and lower indexes, according to:

dT^2 = g_UV dx^U dx^V =dx^U dx_U
= dx^u dx_u + dx^5 dx_5 = dtau^2 + dx^5 dx_5 ? (6)

9) Then just as we can perform ordinary rotations through the D=3 space
dimensions, is there anything which would bar us from considering a
rotation between the usual time dimension x^0 and an axial time
dimension x^5? Then, just as Feynman taught that particles, e.g.,
electrons can move "backwards" through time, might we also "define" the
reference frame of an observer as one in which dx_5=0, always, and
consider some particles as moving with a "sideways" component through
time, dx^5 not==0 relative an observer's movement through time? For
example, might we even think of a massless photon or graviton, for which
dtau=0, as moving fully sideways through time, with dT = dx^5 but
dtau=0?

10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?

Thanks,

Jay.
_____________________________
Jay R. Yablon
Email:

Jay R. Yablon wrote:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv =
n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?


"Ken S. Tucker" wrote in message
ups.com...

IMO, the presences of matter, that reacts with the test
particle used to determine the "field" causes an apparent
"Non-Orthogonal" geometric relation on the trajectory of
the test particle, specifically, deviant from a cartesian
trajectory, and therefore the "weak field" correction "h^uv"
is applicable.
I'll add that the "h^uv" appears as a 1st order perturbation,
and in stronger fields, for example those encountered
within the nucleus itself, more accuruate terms would be
required, and more careful formula.

Actually, Ken, my understanding is that

g^uv = n^uv + kappa h^uv (1)

is *not* a first order equation, but that the deviation from the n^uv
captured by the kappa h^uv applies even in *strong* gravitational
fields. For example, Schwarzschild, an exact solution to the non-linear
field equations, which that applies to strong gravitational fields
equally with weak ones, tells us that:

g_00 = 1 - 2GM/rc^2. (2)

Thus, in this case:

kappa h^00 = - 2GM/rc^2. (3)

Is this correct, or am I overlooking something?

Jay.

Jay R. Yablon wrote:

Jay R. Yablon wrote:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv =
n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?


"Ken S. Tucker" wrote in message
ups.com...

IMO, the presences of matter, that reacts with the test
particle used to determine the "field" causes an apparent
"Non-Orthogonal" geometric relation on the trajectory of
the test particle, specifically, deviant from a cartesian
trajectory, and therefore the "weak field" correction "h^uv"
is applicable.
I'll add that the "h^uv" appears as a 1st order perturbation,
and in stronger fields, for example those encountered
within the nucleus itself, more accuruate terms would be
required, and more careful formula.

Actually, Ken, my understanding is that

g^uv = n^uv + kappa h^uv (1)

is *not* a first order equation, but that the deviation from the n^uv
captured by the kappa h^uv applies even in *strong* gravitational
fields.

We both understand "g^uv" transforms like a tensor,

g'^uv = (&x'^u/&x^a) (&x'^v/&x^b) g^ab,

as does the Kronecker Delta, d^u_v, which is
required to be 0 or 1 in all CS's.

I'm worried, the values you set for the so-called Minkowski
tensor, "n^uv" , (1,-1,-1,-1), necessarily require the Rieman
Christoffel curvature tensor R_abcd =0 to transform as a
tensor. It's perfectly appropriate to use "n^uv" as a tensor
in SR, where of course R_abcd=0, and metric values are
constant, however I hesitate to call "n^uv" a tensor in GR.

If you wish "kappa h^uv" to be a tensor in GR it follows
"n^uv" must be as well, so I'll ask, is it?

For example, Schwarzschild, an exact solution to the non-linear
field equations, which that applies to strong gravitational fields
equally with weak ones, tells us that:

g_00 = 1 - 2GM/rc^2. (2)

Thus, in this case:

kappa h^00 = - 2GM/rc^2. (3)

Is this correct, or am I overlooking something?
Jay.

That solution applies only to the idealistic case of
a point in a vacuum, (G_uv =T_uv =0).

BTW, Weinberg in Grav & Cosmo" pg.212 discusses
the "Post Newtonian Approximation" that is like your
Eq.(1), if you take a moment to glance at it.

Best Regards
Ken

"Jay R. Yablon" wrote in message
...
|
| Jay R. Yablon wrote:
|
| 1) Am I correct to understand that when equation (1) is written with
| g^uv and not with n^uv, that this is done deliberately with the
| understanding that g_uv is the contravariant metric tensor g^uv =
| n^uv +
| kappa h^uv, where h^uv represents a non-zero gravitational field?
|
|
| "Ken S. Tucker" wrote in message
| ups.com...
|
| IMO, the presences of matter, that reacts with the test
| particle used to determine the "field" causes an apparent
| "Non-Orthogonal" geometric relation on the trajectory of
| the test particle, specifically, deviant from a cartesian
| trajectory, and therefore the "weak field" correction "h^uv"
| is applicable.
| I'll add that the "h^uv" appears as a 1st order perturbation,
| and in stronger fields, for example those encountered
| within the nucleus itself, more accuruate terms would be
| required, and more careful formula.
|
| Actually, Ken, my understanding is that
|
| g^uv = n^uv + kappa h^uv (1)
|
| is *not* a first order equation, but that the deviation from the n^uv
| captured by the kappa h^uv applies even in *strong* gravitational
| fields. For example, Schwarzschild, an exact solution to the non-linear
| field equations, which that applies to strong gravitational fields
| equally with weak ones, tells us that:
|
| g_00 = 1 - 2GM/rc^2. (2)
|
| Thus, in this case:
|
| kappa h^00 = - 2GM/rc^2. (3)
|
| Is this correct, or am I overlooking something?

No, and yes.
You overlooked several things, but the most important
is this blunder at the beginning:
http://www.androcles01.pwp.blueyonde...minoEffect.GIF
Androcles.

Ken S. Tucker wrote:
Jay R. Yablon wrote:

Jay R. Yablon wrote:

1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv is the contravariant metric tensor g^uv =
n^uv +
kappa h^uv, where h^uv represents a non-zero gravitational field?


"Ken S. Tucker" wrote in message
ups.com...

IMO, the presences of matter, that reacts with the test
particle used to determine the "field" causes an apparent
"Non-Orthogonal" geometric relation on the trajectory of
the test particle, specifically, deviant from a cartesian
trajectory, and therefore the "weak field" correction "h^uv"
is applicable.
I'll add that the "h^uv" appears as a 1st order perturbation,
and in stronger fields, for example those encountered
within the nucleus itself, more accuruate terms would be
required, and more careful formula.

Actually, Ken, my understanding is that

g^uv = n^uv + kappa h^uv (1)

Just to add a bit,
I've looked at making

g^uv = Orthogonal ^uv + Non-Orthogonal ^uv

where the Orthogonal satisfies R_abcd=0,
as the "n^uv" do. I wasn't successful doing
that yet.

As my studies progressed, I found the metric
expressed, (non-symmetrically),

g_uv = s_uv + a_uv,

was well behaved, and yes I know the asymmetrical
"a_uv" is so unpopular it's voted "crack-pot".

By physically setting a_uv == q*F_uv , which is
a relation between a test particle and field, the
"a_uv" are real, unlike a field "F_uv" which is imaginary.
That's an important distinction given the relativity of
charge, and further provides a means of expressing
mass "m" electromagnetically, (m = q1*q2/r).

We know from experiment, that a charge "q" that varies
it's energy in a field "F_uv" does that by emitting or
acquiring a quanta aka photon, so reality tells us that
"a_uv" is discontinuous, but not constant.

OTOH, Maxwell's eqs. allow a partial derivative like,

& (a_uv) / &x^u , ie. = q*&F_01/&t = q*&E/&t,

where the partial diff "&E/&t" is "displacement
current" and predicted Hertzian waves like for TV.

It's a bit complicated, but it makes sense.
Best
Ken