lrudo...@meganet.net wrote:
Glen Ropella asked:
What is that ^ symbol between dx^a and dx^b?
Merle Lefkoff writes:
I thought a tensor is a description of a multi-dimensional space, like
stress and strain (what I'm going through right now).
Roger Critchlow writes:
Working from the context, I'd guess: the tensor
product between the components <i>dx<sup>a</sup></i>
and <i>dx<sup>b</sup></i>of the stress energy tensor
<i>T<sub>ab</sub></i>, but I've never been too sure '
about tensors.
I haven't read the Verlinde paper (and don't intend
to; I find that treating string theoretical physicists
as ignorable crackpots who, alas, have immense institutional
power makes my life much simpler), but my guess about
the "^" symbol is that it's meant to be the "wedge
product" or "exterior product" rather than the general
tensor product--in other words, it's skew-symmetric.
As an example, in old fashioned terms, the tensor
product of two vectors in R3, a = a1x1+a2x2+a3x3
and b = b1x1+b2x2+b3x3, where you can make the indices
of ai and bj be lower and those on x be upper if you
like, is the "dyad" ab = a1b1x1x1+a1b2x1x2+...+a3b3x3x3,
which skew-symmetrizes to a^b = (a1b2-a2b1)x1^x2 +
(a1b3-a3b1)x1^x3 + (a2b3-a3b2)x2^x3 (or maybe, depending
on taste, to that divided by 2 or 3 factorial).
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
