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Minkowski Metric



 
 
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  #1  
Old January 4th 07, 01:29 AM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
Jack Sarfatti
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Posts: 113
Default Minkowski Metric

"The Minkowski metric, unlike the Riemann
metric, by definition is NOT a generally covariant quantity. This is
now conventional wisdom in the history and foundations of
gravitational physics." wrote Z


The above remark is wrong. inkowskiIn fact the Minkowski metric is a
covariant second rank tensor under GCTs in globally flat spacetime and
it is invariant under Lorentz boosts connecting geodesic observers.

Even when you make the split in intrinsically curved space-time

guv(curved) = (Minkowski)uv + huv(curved)

each term on RHS is generally covariant separately.

Consider 1+1 space-time for computational simplicity.
The O(1,1) Lorentz boosts between geodesic inertial observers are

xi' = Li'^ixi

i = 0,1

x0' = #(x0 - @x1)

x1' = #(-@x0 + x1)

c = 1

# = (1 - @^2)^-1/2

In ordinary notation # = gamma, @ = v = relative speed between geodesic
inertial observers

L0'^0 = #

L0'^1 = - #@

L1'^0 = - #@ = L0'^1

L1'^1 = # = L0'^0

(M)i'j' = L^ii'L^jj'(M)ij

where "Minkowski" = M

M00 = 1

M11 = -1

M01 = M10 = 0

M0'0' = L0'^iL0'^jMij

= L0'^0L0'^0M00 + L0'^1L0'^1M11

= L0'^0L0'^0 - L0'^1L0'^1M11

= #^2 - #^2@^2 = +1 = M00

M0'0' = M00

M1'1' = L1'^iL1'^jMij

= L1'^0 L1'^0M00 + L1'^1L1'^1M11

= L1'^0 L1'^0 - L1'^1L1'^1

= #^2@^2 - #^2 = -1 = M11

M1'1' = M11

M0'1' = L0'^iL1'^jMij

= L0'^0L1'^0M00 + L0'^1L1'^1M11

= L0'^0L1'^0 - L0'^1L1'^1

= 0 = M1'0' = M10 = M01

Therefore the canonical form for the Minkowski metric is invariant under
the linear Lorentz boosts connecting geodesic observers.

Furthermore, the Minkowski metric

(Minkowski)uv

under GCT general nonlinear local coordinate transformations Xu'^u is a
covariant 2nd rank symmetric tensor

(Minkowski)u'v' = Xu'^u Xv'^v (Minkowski)uv
  #2  
Old January 5th 07, 01:17 PM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
Chris
external usenet poster
 
Posts: 24
Default Minkowski Metric

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.

"Jack Sarfatti" wrote in message
...
"The Minkowski metric, unlike the Riemann
metric, by definition is NOT a generally covariant quantity. This is
now conventional wisdom in the history and foundations of
gravitational physics." wrote Z


The above remark is wrong. inkowskiIn fact the Minkowski metric is a
covariant second rank tensor under GCTs in globally flat spacetime and it
is invariant under Lorentz boosts connecting geodesic observers.

Even when you make the split in intrinsically curved space-time

guv(curved) = (Minkowski)uv + huv(curved)

each term on RHS is generally covariant separately.

Consider 1+1 space-time for computational simplicity.
The O(1,1) Lorentz boosts between geodesic inertial observers are

xi' = Li'^ixi

i = 0,1

x0' = #(x0 - @x1)

x1' = #(-@x0 + x1)

c = 1

# = (1 - @^2)^-1/2

In ordinary notation # = gamma, @ = v = relative speed between geodesic
inertial observers

L0'^0 = #

L0'^1 = - #@

L1'^0 = - #@ = L0'^1

L1'^1 = # = L0'^0

(M)i'j' = L^ii'L^jj'(M)ij

where "Minkowski" = M

M00 = 1

M11 = -1

M01 = M10 = 0

M0'0' = L0'^iL0'^jMij

= L0'^0L0'^0M00 + L0'^1L0'^1M11

= L0'^0L0'^0 - L0'^1L0'^1M11

= #^2 - #^2@^2 = +1 = M00

M0'0' = M00

M1'1' = L1'^iL1'^jMij

= L1'^0 L1'^0M00 + L1'^1L1'^1M11

= L1'^0 L1'^0 - L1'^1L1'^1

= #^2@^2 - #^2 = -1 = M11

M1'1' = M11

M0'1' = L0'^iL1'^jMij

= L0'^0L1'^0M00 + L0'^1L1'^1M11

= L0'^0L1'^0 - L0'^1L1'^1

= 0 = M1'0' = M10 = M01

Therefore the canonical form for the Minkowski metric is invariant under
the linear Lorentz boosts connecting geodesic observers.

Furthermore, the Minkowski metric

(Minkowski)uv

under GCT general nonlinear local coordinate transformations Xu'^u is a
covariant 2nd rank symmetric tensor

(Minkowski)u'v' = Xu'^u Xv'^v (Minkowski)uv



  #3  
Old January 5th 07, 02:45 PM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
[email protected]
external usenet poster
 
Posts: 43
Default Minkowski Metric


Chris wrote:
I wonder what it means?

covariant? Tensor? Metric?



I never did know. I probably never will.


They are mathematical concepts which are not of fundamental importance
at all. They are but mere tools of confusion very active in the modern
understanding of physics and mathematics.

You aren't missing anything.

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

I don't think so.


I used to think how far the frontiers of mathematics has gone. Now I
realize it is the frontiers of confusion. Those concepts aren't
important now and they never were - always remember Orwell's lesson -
those who control the past determine the future. Those who control the
present determine the past.

  #4  
Old January 5th 07, 04:52 PM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
hanson
external usenet poster
 
Posts: 2,934
Default Minkowski Metric

wrote in message
oups.com...

Chris wrote:
I wonder what it means?
covariant? Tensor? Metric?
I never did know. I probably never will.


[Schoenfeld]
They are mathematical concepts which are not of fundamental
importance at all. They are but mere tools of confusion very
active in the modern understanding of physics and mathematics.

You aren't missing anything.
I used to think how far the frontiers of mathematics has gone.
Now I realize it is the frontiers of confusion. Those concepts
aren't important now and they never were.

[hanson]
VERY WELL SAID, Schoenfeld. Kudos!

[Schoenfeld]
- always remember Orwell's lesson -
those who control the past determine the future.
Those who control the present determine the past.

[hanson]
True, as long as Orwell meant "the peasantry" by "those"
The peasants are perennial like the grass, feeding and
controlling the intelligentsia, which goes with and is
blown away by the winds of time... perennially so too.
ahahahaha... ahahaha


  #8  
Old January 5th 07, 09:20 PM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
Pmb
external usenet poster
 
Posts: 41
Default Minkowski Metric


"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


  #9  
Old January 5th 07, 10:04 PM posted to sci.math,sci.physics,sci.physics.relativity,sci.philosophy.tech,sci.astro
Ken S. Tucker
external usenet poster
 
Posts: 740
Default Minkowski Metric


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

 




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