Dear Friends

---
As you will soon see, I'm not a native speaking English.
I apologize for my poor language and I hope I'm going
to write almost intellegible sentences.
---

The question arise from studying general relativity.

A differentiable manifold is given and, in each one
of its points the tangent space. Obviously the elements
of the holonomic base changes from point to point,
but it seems to me totally non self evident how to
express that changing in an analytic form suitable
for computation of Christoffel symbols.

The only way to express a vector is giving its component
in an agreed upon base and this carries a tautology.

One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors". I see
but to compute the Christoffel symbols, again, I need
straight the functions

vector_base_h_component_k = vector_base_h_component_k (x_1 ... x_n)

Somewhat heuristic presentation of the subject,
often in polar or spherical coordinates, uses
BOTH the the cartesian local base and the holonomic one,
an hopeless trial on an abstract manifold.

Thanks for paying me attention.
Best regard

Imago Mortis

On Jun 4, 4:23 am, Imago Mortis wrote:

One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors". I see
but to compute the Christoffel symbols, again, I need
straight the functions

vector_base_h_component_k = vector_base_h_component_k (x_1 ... x_n)

The derivation of the Christoffel symbols is rather simple but a
little bit tedious in mathematical excise. Starting with a segment of
space or spacetime below as Riemann described it,

ds^2 = g_ij dq^i dq^j

Or

g_ij dq^i/ds dq^j/ds = 1

Where

** ds = actual segment of space
** g_ij = elements of the matrix or the metric
** dq^i = choice of coordinate system

The distance between two points (A and B) can be written as follows.

S = integral[A, B](ds)

Or

S = integral[A, B](sqrt(g_ij dq^i/ds dq^j/ds) ds)

Or

S = integral[A, B](g_ij dq^i/ds dq^j/ds ds)

Where

** g_ij dq^i/ds dq^j/ds = 1

If points A and B are fixed, then we find the Lagrangian as described
below.

L = g_ij dq^i/ds dq^j/ds = 1

What Christoffel did to derive the Christoffel symbols is to apply the
Euler-Lagrange equations below to the Lagrangian above where each
coordinate yields such a geodesic equation.

d(@L/@(dq^i/ds))/dt = @L/@q^i

Where

** @ = partial derivative

The final result can then be written as follows.

d^2q^i/ds^2 = C^n_ij dq^i/ds dq^j/ds

Where

** C^n_ij = connection coefficients

There are many ways to group the connection coefficients. Christoffel
found the most symmetric form. Decades later, Ricci would invent an
operator based on the Christoffel symbols called the covariant
derivative. If Christoffel were to also present other grouping in the
connection coefficients, there would be several different types of the
covariant derivatives.

Taking another covariant derivative to the geodesic equations would
yield the Riemann curvature tensor. Using a different grouping of the
connection coefficients, it would yield a different Riemann curvature
tensor.

See the alchemy in differential geometry? See the absurdity in GR?

Koobee Wublee wrote in message


[snip]

See the alchemy in differential geometry? See the absurdity in GR?

See the brain rot in a retired aerospace engineer?

Dirk Vdm

Imago Mortis wrote:

[...]
The question arise from studying general relativity.

A differentiable manifold is given and, in each one
of its points the tangent space. Obviously the elements
of the holonomic base changes from point to point,
but it seems to me totally non self evident how to
express that changing in an analytic form suitable
for computation of Christoffel symbols.

The only way to express a vector is giving its component
in an agreed upon base and this carries a tautology.

One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors".

That's correct. If you merely know the differentiable manifold
and its tangent space at each point, you can't compute the
Christoffel symbols. The Christoffel symbols are a connection,
and the connection a manifold is not unique.

The Christoffel connection is determined by two additional
conditions: that the torsion is zero, and that the covariant
derivative of the metric is zero. This second condition can
only be formulated if, in addition to your differentiable
manifold, you have a metric.

The first condition, vanishing torsion, is equivalent to the
requirement that two covariant derivatives commute when
acting on *functions*. In a holonomic (i.e., coordinate) basis,
this implies that the Christoffel connection is symmetric on
its lower two indices. The second condition, metric compatibility,
gives a set of equations for the Christoffel connection that can
be solved uniquely as long as the first condition holds.

The metric compatibility condition is equivalent to saying that
in a (nonholonomic) orthonormal basis, derivatives preserve
orthonormality. In some introductory books, this is assumed
without being explicitly stated, which can lead to confusion.

Steve Carlip

On Jun 4, 12:47 pm, wrote:
Imago Mortis wrote:

The question arise from studying general relativity.
A differentiable manifold is given and, in each one
of its points the tangent space. Obviously the elements
of the holonomic base changes from point to point,
but it seems to me totally non self evident how to
express that changing in an analytic form suitable
for computation of Christoffel symbols.
The only way to express a vector is giving its component
in an agreed upon base and this carries a tautology.
One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors".

That's correct. If you merely know the differentiable manifold
and its tangent space at each point, you can't compute the
Christoffel symbols. The Christoffel symbols are a connection,
and the connection a manifold is not unique.

If you do not know what the metric is at that point in space or
spacetime, you cannot compute the Chritoffel symbols PERIOD.

The Christoffel connection is determined by two additional
conditions: that the torsion is zero, and that the covariant
derivative of the metric is zero. This second condition can
only be formulated if, in addition to your differentiable
manifold, you have a metric.

The covariant derivative is a definition built on top of the
Christoffel symbols. You are getting it wrong here.

The first condition, vanishing torsion, is equivalent to the
requirement that two covariant derivatives commute when
acting on *functions*.

Double covariant derivative gives the Riemann curvature tensor. Ricci
defined that. shrug

In a holonomic (i.e., coordinate) basis,
this implies that the Christoffel connection is symmetric on
its lower two indices.

Christoffel symbols yield a set of symmetric geodesic equations is due
to Christoffel’s own choice of re-arranging these connection
coefficients. shrug

The second condition, metric compatibility,
gives a set of equations for the Christoffel connection that can
be solved uniquely as long as the first condition holds.

You are turning a few manmade mathematical artifacts into something
gospel under the religion of GR. You are creating mysticism.

The metric compatibility condition is equivalent to saying that
in a (nonholonomic) orthonormal basis, derivatives preserve
orthonormality. In some introductory books, this is assumed
without being explicitly stated, which can lead to confusion.

The confusion is your intention. Your (plural) teaching is absolutely
Orwellian in brain-washing effectiveness where

** MYSTICISM IS WISDOM
** PLAGIARISM IS CREATIVITY
** CONJECTURE IS REALITY
** FAITH IS THEORY
** LYING IS TEACHING
** BELIEVING IS LEARNING

On Jun 4, 12:47 pm, wrote:
Imago Mortis wrote:

The question arise from studying general relativity.
A differentiable manifold is given and, in each one
of its points the tangent space. Obviously the elements
of the holonomic base changes from point to point,
but it seems to me totally non self evident how to
express that changing in an analytic form suitable
for computation of Christoffel symbols.
The only way to express a vector is giving its component
in an agreed upon base and this carries a tautology.
One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors".

That's correct. If you merely know the differentiable manifold
and its tangent space at each point, you can't compute the
Christoffel symbols. The Christoffel symbols are a connection,
and the connection a manifold is not unique.

If you do not know what the metric is at that point in space or
spacetime, you cannot compute the Chritoffel symbols PERIOD.

The Christoffel connection is determined by two additional
conditions: that the torsion is zero, and that the covariant
derivative of the metric is zero. This second condition can
only be formulated if, in addition to your differentiable
manifold, you have a metric.

The covariant derivative is a definition built on top of the
Christoffel symbols. You are getting it wrong here.

The first condition, vanishing torsion, is equivalent to the
requirement that two covariant derivatives commute when
acting on *functions*.

Double covariant derivative gives the Riemann curvature tensor. Ricci
defined that. shrug

In a holonomic (i.e., coordinate) basis,
this implies that the Christoffel connection is symmetric on
its lower two indices.

Christoffel symbols yield a set of symmetric geodesic equations is due
to Christoffel’s own choice of re-arranging these connection
coefficients. shrug

The second condition, metric compatibility,
gives a set of equations for the Christoffel connection that can
be solved uniquely as long as the first condition holds.

You are turning a few manmade mathematical artifacts into something
gospel under the religion of GR. You are creating mysticism.

The metric compatibility condition is equivalent to saying that
in a (nonholonomic) orthonormal basis, derivatives preserve
orthonormality. In some introductory books, this is assumed
without being explicitly stated, which can lead to confusion.

The confusion is your intention. Your (plural) teaching is absolutely
Orwellian in brain-washing effectiveness where

** MYSTICISM IS WISDOM
** PLAGIARISM IS CREATIVITY
** CONJECTURE IS REALITY
** FAITH IS THEORY
** LYING IS TEACHING
** BELIEVING IS LEARNING

Why are you academic type so afraid of the truth? Yes, I know in your
power you will procrastinate the nonsense in SR and GR because you
(plural) cannot admit 100 years of absurdity. However, if we can get
to the ones before you poisoned their minds, the truth will prevail.
Since the truth is the most powerful entity in learning, you (plural)
will have no chance in the long run. Professor Carlip and others will
not even be remembered due to their religious belief. shrug

On Jun 4, 12:47 pm, wrote:
Imago Mortis wrote:

The question arise from studying general relativity.
A differentiable manifold is given and, in each one
of its points the tangent space. Obviously the elements
of the holonomic base changes from point to point,
but it seems to me totally non self evident how to
express that changing in an analytic form suitable
for computation of Christoffel symbols.
The only way to express a vector is giving its component
in an agreed upon base and this carries a tautology.
One may say: "The Christoffel symbols are related with
derivatives of base vectors and these can be, non doubt,
represented as linear combination of base vectors".

That's correct. If you merely know the differentiable manifold
and its tangent space at each point, you can't compute the
Christoffel symbols. The Christoffel symbols are a connection,
and the connection a manifold is not unique.

If you do not know what the metric is at that point in space or
spacetime, you cannot compute the Chritoffel symbols PERIOD.

The Christoffel connection is determined by two additional
conditions: that the torsion is zero, and that the covariant
derivative of the metric is zero. This second condition can
only be formulated if, in addition to your differentiable
manifold, you have a metric.

The covariant derivative is a definition built on top of the
Christoffel symbols. You are getting it wrong here.

The first condition, vanishing torsion, is equivalent to the
requirement that two covariant derivatives commute when
acting on *functions*.

Double covariant derivative gives the Riemann curvature tensor. Ricci
defined that. shrug

In a holonomic (i.e., coordinate) basis,
this implies that the Christoffel connection is symmetric on
its lower two indices.

Christoffel symbols yield a set of symmetric geodesic equations is due
to Christoffel’s own choice of re-arranging these connection
coefficients. shrug

The second condition, metric compatibility,
gives a set of equations for the Christoffel connection that can
be solved uniquely as long as the first condition holds.

You are turning a few manmade mathematical artifacts into something
gospel under the religion of GR. You are creating mysticism.

The metric compatibility condition is equivalent to saying that
in a (nonholonomic) orthonormal basis, derivatives preserve
orthonormality. In some introductory books, this is assumed
without being explicitly stated, which can lead to confusion.

The confusion is your intention. Your (plural) teaching is absolutely
Orwellian in brain-washing effectiveness where

** MYSTICISM IS WISDOM
** PLAGIARISM IS CREATIVITY
** CONJECTURE IS REALITY
** FAITH IS THEORY
** LYING IS TEACHING
** BELIEVING IS LEARNING

Why are you academic type so afraid of the truth? Yes, I know in your
power you will procrastinate the nonsense in SR and GR because you
(plural) cannot admit 100 years of absurdity. However, if we can get
to the ones before you poisoned their minds, the truth will prevail.
Since the truth is the most powerful entity in learning, you (plural)
will have no chance in the long run. Professor Carlip and others will
not even be remembered due to their religious belief. shrug

Did anyone notice how Professor Roberts tried miserably to extend the
absurdity of GR by equating a matrix to a scalar below?

http://groups.google.com/group/sci.p...ecb59870?hl=en

I can always second-guess your tactics in dealing with the truth. The
best strategy if I were in your shoes is to just ignore the truth
without even mentioning it. Let’s see how long your insult on ones’
intelligence is going to last.

carlip-nospam wrote on Wed, 04 Jun 2008 19:47:54 +0000:

Imago Mortis wrote:

[...]
The question arise from studying general relativity.

A differentiable manifold is given and, in each one of its points the
tangent space. Obviously the elements of the holonomic base changes
from point to point, but it seems to me totally non self evident how to
express that changing in an analytic form suitable for computation of
Christoffel symbols.

The only way to express a vector is giving its component in an agreed
upon base and this carries a tautology.

One may say: "The Christoffel symbols are related with derivatives of
base vectors and these can be, non doubt, represented as linear
combination of base vectors".

That's correct. If you merely know the differentiable manifold and its
tangent space at each point, you can't compute the Christoffel symbols.
The Christoffel symbols are a connection

They are not the connection in presence of torsion.

The Christoffel connection is determined by two additional conditions:
that the torsion is zero, and that the covariant derivative of the
metric is zero. This second condition can only be formulated if, in
addition to your differentiable manifold, you have a metric.

But the Christoffel *symbols* (OP question) are well defined in presence
of torsion.

The first condition, vanishing torsion, is equivalent to the requirement
that two covariant derivatives commute when acting on *functions*. In a
holonomic (i.e., coordinate) basis, this implies that the Christoffel
connection is symmetric on its lower two indices. The second condition,
metric compatibility, gives a set of equations for the Christoffel
connection that can be solved uniquely as long as the first condition
holds.

But the Christoffel *symbols* (OP question) are well defined when the
connection is *not* symmetric.



--
Center for CANONICAL |SCIENCE)
http://canonicalscience.org

Dear Friends

---
As you will soon see, I'm not a native speaking English.
I apologize for my poor language and I hope I'm going
to write almost intellegible sentences.
---

Thanks you all for your replies !!

I have read them and may be I moved a step further
in understanding the subject:

The matters is similar what happens in topology:
you can't establish if a set is open or not
put aside listing the open subsets because
the list of open subsets IS the topology.

Analogously Christoffel's symbols ARE, in their
very own nature, the connection and computing them
is meaningful only when, a priori, the existence
and compatibility with the metric of the connection
is known.

Hope I wrote a passable approximation of the truth.

Warmest regards !

Imago Mortis