Thus if we are to express GR in terms of the language of QM we must make the radical assumption that the displacement vectors in Minkowski space are pulses of 4-space
which can be expressed in terms of Fourier functions as follows:
Wang, "The Lorentzian surfaces in semi-Euclidean 4-space
with index 2," Journal of Mathematical Analysis and Applications, vol.
The Minkowski 4-space
[E.sup.4.sub.1] is called a Minkowski spacetime.
This is--in contrast to the Lorentz transformation--not a rotation in 4-space
but a linear transformation (stretching) with a translation.
However, by the end of this important book, the diligent reader will be viewing the world of physical phenomena in a more complete 4-space
frame of reference.
The project, known as 4-SPACE
(4 stroke powered gasoline auto-ignition controlled combustion engine), was completed in December after 36 months of work by six partners from three EU Member States: IFP (Institut francais du petrole), Brunel University, GIE PSA Peugeot Citronn, Ruprecht-Karls-Universitat Heidelberg, Ford Motor Company Ltd and DaimlerChrysler AG.The project cost Euro 4.6 million, half of which was provided by the European Commission under the BRITE/EURAM 3 programme of the Fourth Framework Programme of Research and Technological Development.
[D.sub.4] is one such lattice, and, hence, as an approximation of Euclidean 4-space
is somewhat pathological.
But you would have to be a 4-D creature living in 4-space
to see and grasp a 3-torus as easily as you understand the shape of the doughnut sitting next to your coffee.
He tries to alleviate this craziness by comparing it to particle pair creation and annihilation--an analogy which we have seen to be altogether inadequate and in any case irrelevant to the Vilenkin model as Smith interprets it, since he supposedly lacks either the embedding quantum mechanical spacetime or the Euclidean 4-space
. The principle ex nihilo nihil fit seems to me to be a sort of metaphysical first principle, one of the most obvious truths we intuit when we reflect seriously.
It follows that the 4-space
U: = <[C.sub.2], x, [x.sub.13]) contains X[[x.sub.13], [x.sub.24]], [C.sub.2] and [C.sub.3].
The Minkowski space-time [E.sup.4.sub.1] is the Euclidean 4-space
[E.sup.4 equipped with indefinite flat metric given by
1 wherein our familiar 4-space
has a dual 4-space
partner to comprise a unique 8-space.