Minkowski Metric

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

Pmb wrote:
"Chris" wrote in message
o.uk...
I wonder what it means?

covariant? Tensor? Metric?

I never did know. I probably never will.

Will it get me to alpha centuri or make me immortal?

I don't think so.

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

Good luck

Pete

Back in 1983, the International Standards of Units
agreed to set,
Length (n meters) = c Time (second),
with "n" a fixed scalar.

Did you (Pete and anyone else) agree with that
decision or have any input into that decision?
Ken

Nope. I certainly had no input back then. I was as freshman in college at
that time.
Pete

Pmb wrote:

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

If (dq^i dq^j =/= dq'^i dq'^j), then your (19) is wrong.

Therefore, g_ij cannot be a tensor.

With g_ij not being a tensor, it does not invalidate the Christoffel
symbols of the second kind or the Riemann curvature tensor. This only
means the Riemann curvature tensor is not a tensor but a matrix
dependent on the choice of coordinate system. The Ricci tensor is the
same way. Thus, the field equations are merely a set of ordinary
differential equations only valid for the particular choice of
coordinate system with g_ij as the metric to interpret the curvature in
spacetime where the geometry of spacetime is truly invariant.

Pmb wrote:
"Ken S. Tucker" wrote in message
oups.com...

Pmb wrote:
"Chris" wrote in message
o.uk...
I wonder what it means?

covariant? Tensor? Metric?

I never did know. I probably never will.

Will it get me to alpha centuri or make me immortal?

I don't think so.

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

Good luck

Pete

Back in 1983, the International Standards of Units
agreed to set,
Length (n meters) = c Time (second),
with "n" a fixed scalar.

Did you (Pete and anyone else) agree with that
decision or have any input into that decision?
Ken

Nope. I certainly had no input back then. I was as freshman in college at
that time.
Pete

See if this looks ok...
Since 1983, N= 299,792,458
N meters = c * 1 second.
See
http://en.wikipedia.org/wiki/Metre

IMO, that's had a profound effect on Relativity Theory.

In the MKS system of units we have the basis vectors
e1, e2, e3 and e4 such that the magnitudes are

|ei| = 1 meter, {i=1,2,3} and |e4| = c*1 second.

These are related by N*|ei| = |e4| as per definition,
and I'm ok with that International definition and have
learned to live with it just fine.
With it understood the e_u are the 4D basis vectors
and we define our metrics by the scalar products

g_uv = e_u . e_v and g^uv = e^u . e^v

(covariant (cov) and contravariant (con) respectively).

It's easy to see from above (either cov or con)

N*N*g11 = g44 , IOW's,

N^2 Meter^2 = c^2 Second^2.

Those equations are arrived at by applying the
International definition of the MKS system.

To simplify, I'll change the length and time units
so that N=1 and c=1 to get,

g11 = g44

in "flat space".

Using a cartesian CS, we now have,

g11 = g22 = g33 = g44 = 1.

We recover the spacetime interval

ds^2 = dx4 dx4 - dx1 dx1 - dx2 dx2 - dx3 dx3

from

ds^2 = g_uv dx^u dx^v

by employing g_i4 = -g_ij dx^j /dx^4, to yield,

ds^2 = g_44 dx^4 dx^4 - g_ij dx^i dx^j.

Pete, maybe I should write a paper on that,
because it's really involved after that last step.
Regards
Ken S. Tucker

"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Remember the christoffel symbols well before they chopped me brain out "I'm
Chris".

The doctors are nuts.

Chris.

"Pmb" wrote in message
. ..

"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Pmb wrote:
"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick look at
what tensors are. Then you'll be on your way to understanding the terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Hi Pete, I read over your article twice and it looks
good to me, (classical of course). Certainly your
Eq. (19) is perfect. I think KW might be jerking ya.
Regards
Ken

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

Pmb wrote:
"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick look
at
what tensors are. Then you'll be on your way to understanding the
terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Hi Pete, I read over your article twice and it looks
good to me, (classical of course). Certainly your
Eq. (19) is perfect. I think KW might be jerking ya.
Regards
Ken

Hmm ... I don't see the humor in it. - Pete

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

Pmb wrote:
"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick look
at
what tensors are. Then you'll be on your way to understanding the
terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Hi Pete, I read over your article twice and it looks
good to me, (classical of course). Certainly your
Eq. (19) is perfect. I think KW might be jerking ya.
Regards
Ken

Hmm ... I don't see the humor in it. - Pete

Usually KW is a fair guy, I sure hope he didn't
deliberately misquote your carefully crafted
article. Let's wait for him to chime in, maybe
there's a misunderstanding.
Ken

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

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

Pmb wrote:
"Koobee Wublee" wrote in message
ups.com...
Pmb wrote:

To familiarize yourself with these items you should take a quick
look
at
what tensors are. Then you'll be on your way to understanding the
terms
you
seek. See my web page at

http://www.geocities.com/physics_wor...tro_tensor.htm

If anything is unclear then please let me know.

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Pete

Hi Pete, I read over your article twice and it looks
good to me, (classical of course). Certainly your
Eq. (19) is perfect. I think KW might be jerking ya.
Regards
Ken

Hmm ... I don't see the humor in it. - Pete

Usually KW is a fair guy, ..

You're right on that part. He seems like a nice person to me.

...I sure hope he didn't
deliberately misquote your carefully crafted
article. Let's wait for him to chime in, maybe
there's a misunderstanding.
Ken

I think he must have have simply missed it for whatever reason. I don't
seriously think he and I disagree on the math. One of us made an error and I
don't believe the error was mine. But we all do it at one point or another.
Who knows, maybe I'm wrong here too?

Pete

Pmb wrote:
"Koobee Wublee" wrote in message

How about something that is wrong?

In (17) and (18), you have

ds^2 = g_ij dq^i dq^j = g'_ij dq'^i dq'^j

In (19), you have

g_ij = g'_ij

19 does not say that. Please reread.

Yes, you are very correct. It is my mistake. What you are saying is
what I have been telling Mr. Bielawski, Dr. Roberts, Mr. McCollough,
and their sidekicks all along. I must be terribly intoxicated then.

However, I still have a couple questions I would like you to address
that is if you care to do so.

** Why do you call the metric a tensor since the metric is dependent
on the choice of coordinate system? The metric is not invariant while
ds^2 is invariant, and you call it a tensor.

** Why do you think the Riemann curvature tensor is a tensor while the
Christoffel symbols of the second kind are not tensors? The Riemann
curvature tensor is a function of the Christoffel symbols of the second
kind.