Problem with Schutz's Text...

BBaker
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Topic 192385

My name is Bradley Baker and I am a fellow einstein@home user as well as a physics and math teacher at the high school level. I am currently studying General Relativity on my own and was referred to a textbook by Bernard Schutz called "A First Course in General Relativity." And, while it is a very good textbook so far as I can tell, I do have a question.

On page 14 of the paperback edition he states that events in which �delta s squared" is positive are said to be spacelike separated and that events in which that same quantity is negative are said to be timelike separated. I have found this to be in apparent conflict with other texts and sources. "Delta s squared" is the interval between any 2 events separated by the coordinate increments "delta t", "delta x", "delta y" and "delta z". These other sources are basically "vice-verse" on the timelike-spacelike separation issue. The section deals with the invariance of the interval in Special Relativity.

Any light you may be able to shed would be a tremendous help.

Chipper Q
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Problem with Schutz's Text...

Hi Bradley,

I'm trying to learn and understand GR too. Hopefully this is correct: The use of signs in the metric is by choice, the purpose being to keep time different from space. For each pair of tangent vectors v,w at the same point, the metric g(v,w) is like a “dot product� of them. In (easier) flat Minkowski space (of Special Relativity) the vectors look like:

v = (t,x,y,z)
w = (t',x',y',z')

and so then the metric is

g(v,w) = -tt' + xx' + yy' + zz' ,

for the general relativists, so as to keep the geometry of space similar as possible, while the particle physicists would express it as

g(v,w) = tt' – xx' – yy' – zz' .

I learned this from a web-based tutorial that recommends the book you have (along with all the classics). See The General Relativity Tutorial by John Baez. Hopefully you'll find it very helpful :)

Mike Hewson
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RE: My name is Bradley

Quote:

My name is Bradley Baker and I am a fellow einstein@home user as well as a physics and math teacher at the high school level. I am currently studying General Relativity on my own and was referred to a textbook by Bernard Schutz called "A First Course in General Relativity." And, while it is a very good textbook so far as I can tell, I do have a question.

On page 14 of the paperback edition he states that events in which �delta s squared" is positive are said to be spacelike separated and that events in which that same quantity is negative are said to be timelike separated. I have found this to be in apparent conflict with other texts and sources. "Delta s squared" is the interval between any 2 events separated by the coordinate increments "delta t", "delta x", "delta y" and "delta z". These other sources are basically "vice-verse" on the timelike-spacelike separation issue. The section deals with the invariance of the interval in Special Relativity.

Any light you may be able to shed would be a tremendous help.


Yeah, there's two camps, each with their own standard, who throw bolts of tachyons at each other.... :-)
The metric can have either sign, and it all comes out in the wash when the numbers get crunched.
The main thing to keep clear in your mind is those events that are within a light cone's umbrella, versus those that aren't. Beware that this isn't always a 45 degree slope ( if c=1 ), but you can get 'fluted' cones. In any case it is the integration of the metric using those differentials as above that matters.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

BBaker
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RE: RE: My name is

Message 60364 in response to message 60363

Quote:
Quote:

My name is Bradley Baker and I am a fellow einstein@home user as well as a physics and math teacher at the high school level. I am currently studying General Relativity on my own and was referred to a textbook by Bernard Schutz called "A First Course in General Relativity." And, while it is a very good textbook so far as I can tell, I do have a question.

On page 14 of the paperback edition he states that events in which �delta s squared" is positive are said to be spacelike separated and that events in which that same quantity is negative are said to be timelike separated. I have found this to be in apparent conflict with other texts and sources. "Delta s squared" is the interval between any 2 events separated by the coordinate increments "delta t", "delta x", "delta y" and "delta z". These other sources are basically "vice-verse" on the timelike-spacelike separation issue. The section deals with the invariance of the interval in Special Relativity.

Any light you may be able to shed would be a tremendous help.


Yeah, there's two camps, each with their own standard, who throw bolts of tachyons at each other.... :-)
The metric can have either sign, and it all comes out in the wash when the numbers get crunched.
The main thing to keep clear in your mind is those events that are within a light cone's umbrella, versus those that aren't. Beware that this isn't always a 45 degree slope ( if c=1 ), but you can get 'fluted' cones. In any case it is the integration of the metric using those differentials as above that matters.

Cheers, Mike.


Does that mean the bolts of tachyons get there before they are thrown?

Odysseus
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RE: Does that mean the

Message 60365 in response to message 60364

Quote:
Does that mean the bolts of tachyons get there before they are thrown?


And what if your light cone is a Klein bottle?

Mike Hewson
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In a Klein bottle, I think

In a Klein bottle, I think you get hit in the back of the head before you've thrown it. That way it's easier to you remember that you have to do it!! :-)

You should read some of Terry Pratchett's 'Discworld' series. I think in the 'Mort' story the main character's father grows 're-annual' grapes. These are harvested the year before they are grown. It makes for a hangover before you've drunk the wine, amongst other issues. :-)

My ( half-assed ) joke about tachyons was simply to suggest that the choice of differential metric sign convention doesn't create acausal/superluminal conundrums.

One author said 'there is my way, and then there is the crazy way' .... :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Odysseus
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RE: You should read some of

Message 60367 in response to message 60366

Quote:
You should read some of Terry Pratchett's 'Discworld' series. […]


Oh, I have done, most of them. There’s quite a bit of quasi-scientific humour there, amidst the swords-&-sorcery parody and the Astérix et Obélix-esque ethnographic caricatures: a very lively wit. His non-Discworld stuff, like Only You Can Save Mankind, is also excellent.

Chipper Q
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RE: RE: Does that mean

Message 60368 in response to message 60365

Quote:
Quote:
Does that mean the bolts of tachyons get there before they are thrown?

And what if your light cone is a Klein bottle?


Wow... hmm... might something like that mean that black holes never quite merge, but instead get knotted up in Klein-bottle-shaped distortions in spacetime? Could several (just a few of average mass), say in the center of a galaxy, get so entangled that spacetime gets pulled into a topology that is equivalent to millions of black holes (e.g., a kind of tangled superposition of the local spacetime, as the black holes go repeatedly in and out/through each other's Klein bottles)? That'd be a scary model... :)

ragnar schroder
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RE: I'm trying to learn

Message 60369 in response to message 60362

Quote:

I'm trying to learn and understand GR too.

I went the same route of studying GR too, using mainly the Baez website and various books, including the one mentioned.

I'm left with the feeling the theory is formulated "too elegantly", hiding away the actual physics behind a layer of abstraction.

In the Lagrange/Hamiltonian abstraction of Newtonian physics the basic Newtonian concepts of force, acceleration, speed and position are abstracted away into "generalized coordinates" that work for calculations, but in themselves have no physical meaning.

I have a feeling that the dif.geo. formulation of GR is kind of similar to this - just a seductively elegant way of hiding a more "banal" theory behind mathematical machinery.

Just my two cents - I'm just a dabbler into physics anyway.

Greetings, Mr. Ragnar Schroder

tullio
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RE: In the

Message 60370 in response to message 60369

Quote:
In the Lagrange/Hamiltonian abstraction of Newtonian physics the basic Newtonian concepts of force, acceleration, speed and position are abstracted away into "generalized coordinates" that work for calculations, but in themselves have no physical meaning.


The Hamiltonian is just the sum of kinetic energy and potential energy. This applies to quantum mechanics too, where it becomes an operator, whose eigenfunctions correspond to the energy levels. H(psi)= E(psi) is the basic equation in the static case.
Tullio

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