On 2/9/15 2/9/15 - 11:33 AM, Jack...@hotmail.com
> On Sun, 8 Feb 2015 00:14:56 CST, Tom Roberts
>> The key point is that semi-Riemannian geometry is NOT Riemannian
>> geometry. While many theorems of the latter apply to the former,
>> not all do, and often they get a minus sign or absolute-value.
>> And for many purposes, one must treat spacelike and timelike
>> intervals separately (signs often differ).
> I'm saying that I can't believe in general relativity, because even
> just the mathematical basis is defective.
It is YOU who is mistaken, not GR which is "defective".
This is not really about "belief", it is about UNDERSTANDING, which is more
subtle than you appreciate.
> If it was valid mathematics,
> you would have a "metric tensor" that would bend space-time, in which
> dtau/dt = g00 = sqrt(1-2MG/rc^2). instead you use the negative square
> of that term.
Look at the quote up above -- Semi-Riemannian geometry is NOT Riemannian
geometry. You are attempting to force GR into the latter, and that is utterly
[I ignore the difference betwen metric signatures -+++ and +---.]
> implicit, in such a case, would be an obligation to specifically list
> the basis vectors for that space, but that can't be done without
> listing ict.
Sure it can. Indeed "ict" is long gone from GR. As you mention, there is even a
box in MTW that explains why -- "ict" was an attempt to force GR into Riemannian
geometry; like your attempt above, it failed (the introduction of complex
quantities causes various theorems to fail, just as they fail in semi-Riemannian
Select a locally inertial frame at a given point in the
spacetime manifold of interest. The basis vectors are:
(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)
The metric is diag(-1,1,1,1). There's not an "i" involved.
Again, your complaint is really about your personal ignorance, not about GR itself.
> I point out the only case known to me, of an overt attempt at such
> expurgation is in "Gravitation" page 51, by MTW, titled "Farewell to
> ICT". It is a pathetic explanation, something like not wanting X4,
> with its ict, to confuse people and therefore selecting X0 as the time
> Included is some twaddle about vectors being entirely different
> creatures from one forms, when you really only need covariant and
> contravariant forms of vector corresponding to row and column
> I'm not sure of the exact impact of making that discrimination.
Hmmmm. Vectors _ARE_ different from one-forms; vectors and one-forms _ARE_ the
same as contravariant vectors and covariant vectors, and they _ARE_ also the
same as vectors and covectors.
(Beware the meaning of "co-", as it can mean either "with" or
"against/inverse". In some contexts the meanings of "covariant"
and "contravariant" are inverted. Remember also that a covariant
vector has a set of contravariant components; the confusion
multiplies when tensors are confused with their components.
But still, a co-mathematician would be a system for converting
theorems into coffee.)
Note that vectors do NOT "correspond" to column matrices. You are confusing the
COMPONENTS of the vector with the vector itself. That's invalid (even though
many ancient books confuse them). A vector has a magnitude and a direction. To
turn a column matrix into a vector requires applying the corresponding set of
basis vectors -- these last are indeed VECTORS, and not merely another column
> Quadratic form should be of the form s^2 = X'GX or x^i*x_jGij but
> definitely not G*(x^i)^2.
I have no idea what you are trying to say. Every quadratic form I know of in GR
(and in differential geometry) can be written as A(X,X), for a rank-2 tensor
A(.,.) and a vector X; in components that would be A_ij X^i X^j.
Note all three of your formulas are wrong, in one detail or
another. For someone complaining about lack of rigor that is
> I bow to your superior knowledge of general relativity. I'm just
> telling you that there's so much that is hard to believe in, given the
> slovenly treatment of mathematics.
You need a better textbook. Physicists are well known for their casual attitude
about mathematical rigor. You also need to improve the precision of your own
thoughts and words, as YOU are excessively casual, too.
> I know enough real physics to know that time is not dilated; the
> effects merely describe the change in the velocity of light in
What you think you know is wrong. The local vacuum speed of light DOES NOT
CHANGE, in gravity or anywhere else. When you omit the qualifier "local" you
merely exhibit your personal ignorance of its importance, demonstrating that you
fail to understand the geometrical structure of GR. Omitting the qualifier
"vacuum" indicates you do not understand the underlying physics.
As I have said so often, it simply is not possible to understand subtle
concepts, such as relativity, without precision in thought and word. Your
statements here fall short of what is needed. Relativity is not only more subtle
than you think, it is more subtle than it is possible for you to think, until
you clean up your act. Complaining about lack of rigor in other peoples'
presentations is the least of your problems.