Exploring Minkowski Metric and Tensors in General Relativity

[therapeuter]

hello. I'm working on a philosophical summary of general relativity. i have difficulty understanding tensor. i made the following characterization; can any expert minds here tell me if i said it correctly?

the Minkowski spacetime metric corresponds to the Kronecker delta (which marks the metric structure of ++++) only because of the use of the imaginary number in the time element, such that the positive d(\|-1(ct))^2 takes up the place of -c^2dt^2. If the negative sign of the time differential were kept, and the metric structure became +++- as it is in its original form, the metric tensor for the Minkowski metric would then be:

$$\left ( \begin {array} {cccc} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \end{array} \right )$$

if anyone can take a peek to see if what i got so far is correct that'd be sooooooo appreciated too:

http://www.theophoretos.hostmatrix.org/generalrelativity.htm

p.s. i also did a special relativity, but it's very long.
http://www.theophoretos.hostmatrix.org/relativity.htm

 

[Los Bobos]

hello. I'm working on a philosophical summary of general relativity. i have difficulty understanding tensor. i made the following characterization; can any expert minds here tell me if i said it correctly?



if anyone can take a peek to see if what i got so far is correct that'd be sooooooo appreciated too:

http://www.theophoretos.hostmatrix.org/generalrelativity.htm

p.s. i also did a special relativity, but it's very long.
http://www.theophoretos.hostmatrix.org/relativity.htm

At least, for me, your text makes very little sense. Why do you want to mess with comples numbers, which have nothing to do with the subject (here we are handling real (psedo)manifolds). In the Minkowski metric the time component has a different sign than the spatial part in order to have the causality needed. And the spacetime invariant is ds^2=dt^2 - dr^2 (or the other way round, just convention). Minkowski metric is not whatever 4-dimensional diagonal matrix.

 

[therapeuter]

LB, are you sure you know about relativity at all? you seem to be speaking gibberish.

does anyone here who knows about relativity have any comment? thanks.

 

[coalquay404]

LB, are you sure you know about relativity at all? you seem to be speaking gibberish.

does anyone here who knows about relativity have any comment? thanks.

I'll comment. Having read the http://www.theophoretos.hostmatrix.org/generalrelativity.htm you linked to, I have to say that it doesn't seem to make much sense to me either. For example, you begin with

General relativity is discovered when special relativity is discovered to be just another "cave" and the ascent is continued. As we ascend further up the Platonic cave, we notice again that what we have then taken as the real things of which what were earlier thought to be the real things were merely shadows, are again merely shadows of some other real things. This constitutes the essence of the story of the ascent from special relativity to general relativity. After we have realized that what have seemed to be the invariant space distances and time intervals are merely shadows -- images, likeness (eikasia) -- cast by the "real" Minkowski spacetime metric $ds^2 = (dx^1)^2 + (dx^2)^2 + (dx^3)^2 + (dx^4)^2$ which is the real invariant "interval" (under Lorentz transformation), we are now, with general relativity, coming to the realization that this Minkowski metric itself is only a shadow of the Gaussian spacetime metric.

Laboured rhetorical flourishes aside, this is completely incorrect. Firstly, the base Minkowski line interval is $ds^2 = \pm((dx^1)^2 + (dx^2)^2 + (dx^3)^2 - (dx^4)^2)$, with the $\pm$ being simply a matter of convention. Secondly, what is a "Gaussian spacetime metric"?

Next, you say:

What does "shadow" mean here? It means that, while the Minkowski spacetime metric is valid for an Euclidean spacetime, the "real" spacetime -- spacetime on its total scale -- is non-Euclidean, within which the Minkowski metric has only a limited, local, "special" validity: it is valid for an infinitely small region of spacetime that may under all considerations be taken as Euclidean, just as, while the surface of the Earth is non-Euclidean, not flat, but curved, an infinitely small region on it may well for all purposes be considered Euclidean, i.e. appear flat enough.

Again, this is incorrect. The Minkowski metric applies to a Minkowski spacetime, not to a Euclidean spacetime (for which the appropriate metric would be $\delta_{ij}=\mathrm{diag}(++++)$). In a broader context, a general spacetime is said to be Lorentzian, i.e., locally Minkowski, but most certainly not Euclidean.

Finally, Los Bobos' comments about your use of complex numbers in relation to the Minkowski metric are completely justified.

 

[robphy]

You might want to consult these surveys, written by philosophers of science:

http://arxiv.org/abs/gr-qc/0506065
http://www.lps.uci.edu/home/fac-staff/faculty/malament/

(I haven't looked at the following sites in detail)
http://www.tc.umn.edu/~janss011/
http://ls.poly.edu/~jbain/philrel/
http://www.bun.kyoto-u.ac.jp/%7Esuchii/spacetime2004/

 

[Los Bobos]

LB, are you sure you know about relativity at all? you seem to be speaking gibberish.

does anyone here who knows about relativity have any comment? thanks.

Yes I am sure I know about relativity.

 

[therapeuter]

alright the discussion is finished. feel like in a twilight zone or some kind. how can you not even know what Gaussian metric is? and if minkowski metric doesn't apply to euclidean, then i should give you my head. lorentzian? this is so strange. from which world did your guys come from? it's almost like suddenly people start saying, "the sun has always risen from the west". bizarre.

 

[therapeuter]

Moderator, Help, Come Check Out What's Going On Here

 

[robphy]

the Minkowski spacetime metric corresponds to the Kronecker delta

The Minkowski metric, denoted $$g_{ab}$$, maps a pair of vectors to a real number... or a vector to a covector.

The Kronecker delta, denoted $$\delta_a{}^b$$, is simply the identity operator (or index substitution operator), which maps a vector to a vector... or a covector to a covector.. or a vector and a covector to a number, their transvection.

In fact, $$g_{ab}g^{bc}=\delta_a{}^c$$. In matrix language, $$G G^{-1}=I$$

alright the discussion is finished. feel like in a twilight zone or some kind. how can you not even know what Gaussian metric is? and if minkowski metric doesn't apply to euclidean, then i should give you my head. lorentzian? this is so strange. from which world did your guys come from? it's almost like suddenly people start saying, "the sun has always risen from the west". bizarre.

Could it be that you are using nonstandard and possibly archaic terms?
If so, then it might be a good idea to clearly define your terms for your reader. It will not suffice to make a reference to an old paper in a foreign language... primary source or not.

"feel like in a twilight zone or some kind ...from which world did your guys come from"...
In our defense, the world that you have wandered into uses modern differential geometry and tensor analysis. Have a look at the first review paper I referenced.

 

[Chris Hillman]

Suggest Sklar

Therapeuter, I didn't try to follow the link you offered, since I'm not sure what you mean by "philosophical summary" and how seriously you take this. For what it is worth, a very fine nontechnical book you might like is Lawrence Sklar, The Philosophy of Space and Time, which offers a nice introduction to many classical topics, including a chapter on how philosphers have reacted to the appearance of general relativity.

From what you wrote here it seems that you might be trying to talk about some of the same "local versus global" issues that I discussed in connection with "the" equivalence principle in another thread earlier today, but I find it too exhausting to try to explain this rather subtle concept except to an audience comfortable with tensor fields.

Coalquay404: the paragraph you quoted didn't make much sense to me either, possibly because it was taken out of context, but it probably helps to know that the references to "shadows" probably are intended to invoke the so-called "allegory of the cave" due to Plato, which is very well known to undergraduate students of philsophy. (Therepeuter can confirm or deny this the validity of this guess.) Ironically, since we are talking about metrics, the name "Plato" is a nickname or possibly a literary pseudonym which means more or less literally "wide"; some say this moniker probably originated as a schoolboy nickname suggesting a stocky or burly build, or, some say, a chubby one.

Chris Hillman

 

[Daverz]

alright the discussion is finished. feel like in a twilight zone or some kind. how can you not even know what Gaussian metric is? and if minkowski metric doesn't apply to euclidean, then i should give you my head. lorentzian? this is so strange. from which world did your guys come from? it's almost like suddenly people start saying, "the sun has always risen from the west". bizarre.

I've never heard of a "Gaussian metric", either. I checked several standard GR texts and found Gaussian normal coordinates, Gauss-Codazzi equations, Gaussian curvature and, of course, Gauss's Law, but nothing on "Gaussian metric".

Lorentzian is a standard description for a metric with signature like (-+++) (Wald, page 23). Wald is the closest thing to a canonical text in the field, so I would recommend reading the first several chapters of it for the standard terminology.

I think I have seen "Minkowski metric" used for the "Euclideanized" spacetime with t -> ict, usually in old books on special relativity. It's sometimes still used in QFT, though.

 

[Daverz]

The full reference is

Robert M. Wald, General Relativity, University Of Chicago Press, 1984 (ISBN 0226870332)

 

[AlphaNumeric]

A quick use of Google seems to suggest that it's not that you can have a Gaussian metric, but Gaussian fluctuations to a metric (or even non-Gaussian ones). The metric itself is not described as Gaussian.

Perhaps it's just because I'm not one for hardcore philosophy but if someone asked what I thought of describing special and general relativities as caves and images and shadows I'd tell them to actually learn some of relativity because that's nothing like what I'd describe it as.

therapeuter, if you're mixing up Euclidean and Minkowski metrics, I think you might want to slow down and get your head around relativity before trying to summarise it's philosophy to other people who aren't familiar with it. Otherwise it'll be a case of the blind leading the blind by the looks of it.

 

[therapeuter]

i figure out what this twilight zone circus is about now. i know where you guys are from. you have evacuated this forum. you then try to b.s. as real as possible about general relativity, as if you knew relativity and i didn't, and hope to discredit me in front of the important people.

it's not going to work. Gaussian, euclidean = minkowski etc. are very basic concepts. you ain't going to fool anybody but you are only going to expose your stupidity even more and sink further into disrepute. the important people have physicists on their side to verify that i know what I'm talking about here, while you guys don't.

 

[robphy]

Gaussian, euclidean = minkowski etc. are very basic concepts. you ain't going to fool anybody but you are only going to expose your stupidity even more and sink further into disrepute. the important people have physicists on their side to verify that i know what I'm talking about here, while you guys don't.

A more useful and helpful attitude would have been to enlighten us with your definitions.

A full-text search for the phrase "gaussian metric" at JSTOR ( http://www.jstor.org ), yielded 3 results: from 1925, 1942, and 1953. None of them seem relevant, except possibly for the 1953 one, which appears in AmJMath, but is used in a very specific way. ("gaussian spacetime" and "gaussian spacetime metric" yields zero hits. By comparison, "euclidean metric" yields 1058 hits, "riemann metric" yields 2889 hits, "riemannian metric" 3196 hits)


As you requested, I "took a peek" at one of your webpages, I am confused by your explanation of the twin paradox:
In http://www.theophoretos.hostmatrix.org/relativity.htm ,

Just before the References, you say
"The triangle representing spacetime path of the two twins as drawn by the twin remaining on Earth. The elapsed time to the traveling twin (his proper time) is less since the length of the bent path is less than the length of the straight line. "This 'inequality of the triangle of spacetime' is the reverse of the inequality of the triangle of ordinary Euclidean space, for which the sum of the sides is longer than the [straight line]. This difference is due to the particular form of the chronogeometry of spacetime, where the Pythagorean theorem contains a negative sign for the squares of the sides of a right triangle which are directed in time."
which sounds right... but then you say...
"For example, while the twin at rest has traveled 0 - 8 = -8 interval of space time, the traveling twin has traveled (3 - 5) + (3 - 5) = -4 (spacetime distance = spatial distance - time difference)."
Here is your associated diagram:
http://www.theophoretos.hostmatrix.org/spacetimetriangletwin.jpg
which appears to assign Euclidean lengths (i.e. lengths that would be appropriate for a Euclidean space with its Pythagorean theorem) to the legs of the Minkowski-spacetime triangles.

(You assign 5 to the outgoing traveller's leg (the hypotenuse) [and the incoming traveller's leg], 4 to its temporal component (the adjacent side), and 3 to its spatial component (the opposite side), where the angle is taken between the vertical-on-the-spacetime-diagram stay-at-home-twin and the outgoing-traveller leg.)

Using those edge-lengths, the correct assignments [for Special Relativity] are 3 for the outgoing traveller's leg [and the incoming traveller's leg], 5 for its temporal component, and 4 to its spatial component. Then, for the round-trip, the stay-at-home logs 10 units, whereas the traveller logs, via the spacetime-version of the Pythagorean theorem, (sqrt(5^2-4^2)+sqrt(5^2-4^2)=) 6 units.

 

[pervect]

I'm sure then that it will be no trouble for you to cite some examples of usage of "Gaussian metric" then.

I should add that I frequently run into situations in which the terminology varies slightly depending on whether one is talking to a mathematician or a physicist, so some questioning about terminology is not all that uncommon.

Meanwhile, you might want to tone down your personal attacks on some of our more respected members and the forum in general.

 

[AlphaNumeric]

you then try to b.s. as real as possible about general relativity, as if you knew relativity and i didn't, and hope to discredit me in front of the important people.

What important people? And besides, some of us do know a bit about GR.

Gaussian, euclidean = minkowski etc. are very basic concepts. you ain't going to fool anybody but you are only going to expose your stupidity even more and sink further into disrepute. the important people have physicists on their side to verify that i know what I'm talking about here, while you guys don't.

Eh?! There's no need to just turn around and say "No, you're stupid!"

If you are adament that Euclidean = Minkowski you're either working to a TOTALLY different definition to the rest of physics or you're the one 'exposing your stupidity even more'. The two are similar but have s critical difference. Euclidean metrics are positive definite, Minkowski isn't, it's a Lorentzian metric which means it has some positive and some negative eigenvalues, not all the same sign.

Just check Wikipedia and you'll see that that is right. This forum doesn't exist to spread misinformation, it's here to aid people. Why did you bother to ask for advice here if you just say "What?! I'm not wrong, you are!" when someone actually tried to correct you?!

 

[Moonbear]

To prevent this from deteriorating further, I'm locking this thread until one of the physics mentors can review it. They may or may not decide to unlock it again, but things seem to be getting overheated in here to let it continue unmonitored.

 

[chroot]

It's garbage, and should remain locked.

- Warren