Dimensionalities of Space-Time
Yu, Hoi-Lai
16th Oct. 2008
NTHU
Basic Questions in searching the Truths of Nature:
What is Space?
What is Time?
What is the meaning of being somewhere?
What is meaning of “moving”?
Is motion to be defined with respect to objects or with respect to space?
Can we formulate Physics without referring to time or space?
What is matter?
What is causality?
What is the role of Observer in Physics?
Active Diffeormophism invariant
What disappears in GR is precisely the
background space time that Newton believed
to have been able to detect; now the existence of
Gravity is a matter of one’s coordinate choices.
Reality is not made by particles and fields on
Spacetime. Reality is made by particles and fields
(incldue gravity), that can only be localized with
respect to each other.
Spacetime emergence of solution of theory
• Einstein field equation： A theory on space, time
and Matter
R
1
 Rg   kT
2
Geometry of
space time
Quantization
Quantum Gravity？
Spacetime is curved
due to the gravitational
action of matter
Matter
Quantization
QM,QED,QCD…
Puzzles in general relativity
(1)Black holes entropy
A
S
4G
(2)Cosmology: Big Bang
Singularity
Causal Dynamical Triangularization (Lorentzian signature)
Important consequences of CDT:
(1)Baby Universe eliminated
(2)Global Times with respect to foliation
(3) No topology change
(4) New universality class then Euclidean
Four-dimensional Picasso.
Connecting many triangular bits of flat space-time to make curvy
space-times demonstrates how our four-dimensional world can arise
from the fuzzy "quantum foam" that may exist at the tiniest scales.
(Image from researchers who used "non-causal" triangles.)
Causality implies four-dimensionality!
Hausdorff dimension:
V(r) ~ rd
H
Where the gluing for each N4is chosen such that every single building
Block share a given vertex.
Classical spacetime no matter how curved – with finer resolution,
it will always eventually looks like a piece of flat spacetime
Not the case in QG, never resemble a flat space, no matter how fine.
Need a path integral allows for short-scale fluctuation in curvature,
but in such a way the resulting large scale geometry
does not degenerate completely
Causality require that each spacetime appearing in the sum
over geometry should have a specific form, namely object
that can obtain by evolving a purely spatial geometry in time
Volume-Volume Correlator:
Another evidence for spacetime dimension 4:
Spectral dimension
Effective mini-superspace and
Wave Function of Universe (Bounce Solution):
Tunneling of from nothing(zero scale factor a)
From general scaling of the above action it is clear
that the only chance to obtain the observed scaling
law, expressed in terms of the variable t/V1/4 4,
is by setting α = 1/3.
In Euclidean signature and proper time coordinates,
themetrics are of the form
Nonperturbative Quantum Einstein Gravity
The phase portrait of Einstein Hilbert trunction
Now, NGFP predicts:
4 dimension at low energies
2 at high energies
Spectral Dimension
Conclusions:
(1) Space-time dimensiona changes from
2 to 4 when energy resolutions drops
(2) Causality requires a 4 dimensional
Space-time at low energies
(3) Spherical Symmetric Spacetime is
perfered at high energies.
(4) Quantum gravity can be test by gramma
ray bursts of neutron stars.