What's more general than Riemann space?
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From: Igor on 15 Jun 2010 16:43 On Jun 14, 6:49 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > "Igor" <thoov...(a)excite.com> wrote in message > > news:2853b0ef-2239-4ac6-be37-d76e8fce12c7(a)8g2000vbg.googlegroups.com... > On Jun 14, 1:12 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > > > > > "Igor" <thoov...(a)excite.com> wrote in message > > >news:83cc04c2-eb0a-4041-9a36-3eea4365d424(a)a30g2000yqn.googlegroups.com.... > > On Jun 13, 10:49 am, "Androcles" <Headmas...(a)Hogwarts.physics_z> > > wrote: > > > > "Igor" <thoov...(a)excite.com> wrote in message > > > >news:bd35ebdd-08a8-42eb-abf7-614afe8a7eb2(a)e5g2000yqn.googlegroups.com.... > > > On Jun 12, 8:04 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > > > > > "Edward Green" <spamspamsp...(a)netzero.com> wrote in message > > > > >news:4c04f840-cc88-437e-b4e0-ffe904ca73fc(a)35g2000vbj.googlegroups.com... > > > > | What's more general than Riemann space? > > > > > Euclidean space. > > > > BUZZ!!! Euclidean space is a subset of Riemann space. No cookie for > > > you. > > > ================================================= > > > BUZZ!!! > > > Riemann space has a different postulate to Euclidean space. > > >http://en.wikipedia.org/wiki/Parallel_postulate > > > There is NO parallel postulate in Riemann space. You lose. > > > ============================================ > > BUZZ!!! > > Lines of longitude are postulated to be parallel at the equator. > > That's only because a sphere is locally Euclidean. All surfaces are. > ============================================== > Only, huh? > > Non-Euclidean geometries were only generated AFTER the parallel > postulate was removed. > =============================================== > BZZZZZZZZT!!!!!! > > Riemann's parallel postulate: > If a straight line crossing two straight lines makes the interior angles > equal to two right angles, the two straight lines, if extended indefinitely, > eventually MEET. That's only because Riemann examined elliptical geometry where there are NO parallel geodesics to a given line. > Lobachevsky's parallel postulate: > If a straight line crossing two straight lines makes the interior angles > equal to two right angles, the two straight lines, if extended indefinitely, > eventually DIVERGE. And that's only because Lobachevsky studied hyperbolic geometry where there an infinite number of parallel geodesics to a given line. > You can't distinguish Lobachevsky's geometry from Riemann's geometry without > it. But today, all of these are included in the notion of Riemann space. Come into the twenty first century. Apparently, you're still uses words like "Zounds" and "Gadzooks".
From: Androcles on 15 Jun 2010 17:00 "Igor" <thoovler(a)excite.com> wrote in message news:f10c4ad3-639d-44ee-99d9-e7ba8d24768f(a)i28g2000yqa.googlegroups.com... On Jun 14, 6:49 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > "Igor" <thoov...(a)excite.com> wrote in message > > news:2853b0ef-2239-4ac6-be37-d76e8fce12c7(a)8g2000vbg.googlegroups.com... > On Jun 14, 1:12 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > > > > > "Igor" <thoov...(a)excite.com> wrote in message > > >news:83cc04c2-eb0a-4041-9a36-3eea4365d424(a)a30g2000yqn.googlegroups.com... > > On Jun 13, 10:49 am, "Androcles" <Headmas...(a)Hogwarts.physics_z> > > wrote: > > > > "Igor" <thoov...(a)excite.com> wrote in message > > > >news:bd35ebdd-08a8-42eb-abf7-614afe8a7eb2(a)e5g2000yqn.googlegroups.com... > > > On Jun 12, 8:04 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > > > > > "Edward Green" <spamspamsp...(a)netzero.com> wrote in message > > > > >news:4c04f840-cc88-437e-b4e0-ffe904ca73fc(a)35g2000vbj.googlegroups.com... > > > > | What's more general than Riemann space? > > > > > Euclidean space. > > > > BUZZ!!! Euclidean space is a subset of Riemann space. No cookie for > > > you. > > > ================================================= > > > BUZZ!!! > > > Riemann space has a different postulate to Euclidean space. > > >http://en.wikipedia.org/wiki/Parallel_postulate > > > There is NO parallel postulate in Riemann space. You lose. > > > ============================================ > > BUZZ!!! > > Lines of longitude are postulated to be parallel at the equator. > > That's only because a sphere is locally Euclidean. All surfaces are. > ============================================== > Only, huh? > > Non-Euclidean geometries were only generated AFTER the parallel > postulate was removed. > =============================================== > BZZZZZZZZT!!!!!! > > Riemann's parallel postulate: > If a straight line crossing two straight lines makes the interior angles > equal to two right angles, the two straight lines, if extended > indefinitely, > eventually MEET. That's only because Riemann examined elliptical geometry where there are NO parallel geodesics to a given line. > Lobachevsky's parallel postulate: > If a straight line crossing two straight lines makes the interior angles > equal to two right angles, the two straight lines, if extended > indefinitely, > eventually DIVERGE. And that's only because Lobachevsky studied hyperbolic geometry where there an infinite number of parallel geodesics to a given line. > You can't distinguish Lobachevsky's geometry from Riemann's geometry > without > it. But today, all of these are included in the notion of Riemann space. Come into the twenty first century. Apparently, you're still uses words like "Zounds" and "Gadzooks". ================================================= That's only because you are clueless. Oh wait... I forgot to say "Buzz". Very 21st century, that.
From: J. Clarke on 16 Jun 2010 08:38 On 6/12/2010 7:04 PM, Edward Green wrote: > What's more general than Riemann space? Apparently it results from > relaxing the requirement that there be a metric tensor, and is > conventionally denoted by a capital script letter which I cannot > decipher. What is likely the letter, and what is the space called? If you could provide the sentence or paragraph in which this undecipherable letter appears and perhaps the title of the source, it might be easier for someone to help you.
From: Don Stockbauer on 16 Jun 2010 09:20 On Jun 16, 7:38 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > On 6/12/2010 7:04 PM, Edward Green wrote: > > > What's more general than Riemann space? Apparently it results from > > relaxing the requirement that there be a metric tensor, and is > > conventionally denoted by a capital script letter which I cannot > > decipher. What is likely the letter, and what is the space called? > > If you could provide the sentence or paragraph in which this > undecipherable letter appears and perhaps the title of the source, it > might be easier for someone to help you. What's more general than Riemann space? The space of all spaces?
From: Edward Green on 21 Jun 2010 12:55
On Jun 16, 8:38 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > On 6/12/2010 7:04 PM, Edward Green wrote: > > > What's more general than Riemann space? Apparently it results from > > relaxing the requirement that there be a metric tensor, and is > > conventionally denoted by a capital script letter which I cannot > >decipher. What is likely the letter, and what is the space called? > > If you could provide the sentence or paragraph in which this > undecipherable letter appears and perhaps the title of the source, it > might be easier for someone to help you. Sorry for slow response: as I said, D.F. Lawden, "Introduction to Tensor Calculus, Relativity and Cosmology", p.89 "It will be proved in Chapter 6 that, in the presence of a gravitational field, space-time ceases to be Euclidean in Minkowski"s sense and becomes an R_4. This is our chief reason for considering such spaces. However, we can generalize the concept of the space in which our tensors are to be defined yet further. Until section 37 is reached, we shall make no further reference to the metric. This implies that the theory of tensors, as developed thus far, is applicable in a very general N-dimensional space in which it is assumed it is possible to set up a coordinate frame but which is not assumed to possess a metric. In such a hypothetical space, the distance between two points is not even defined. It will be referred to as #_N. R_N is a particular #_N for which a metric is specified." # was since identified as a script S. |