Dimensionalities of Space-TimeDimensionalities of Space-Time

Yu, Hoi-LaiYu, Hoi-Lai

1616thth Oct. 2008 Oct. 2008NTHUNTHU

Basic Questions in searching the Truths of Nature:Basic Questions in searching the Truths of Nature:

What is Space? What is Space?

What is Time?What is Time?

What is the meaning of being somewhere?What is the meaning of being somewhere?

What is meaning of “moving”?What is meaning of “moving”?

Is motion to be defined with respect to objects or with respect to space?Is motion to be defined with respect to objects or with respect to space?

Can we formulate Physics without referring to time or space?Can we formulate Physics without referring to time or space?

What is matter?What is matter?

What is causality?What is causality?

What is the role of Observer in Physics?What is the role of Observer in Physics?

What disappears in GR is precisely the What disappears in GR is precisely the background space time that Newton believed background space time that Newton believed to have been able to detect; now the existence ofto have been able to detect; now the existence ofGravity is a matter of one’s coordinate choices.Gravity is a matter of one’s coordinate choices.Reality is not made by particles and fields on Reality is not made by particles and fields on Spacetime. Spacetime. Reality is made by particles and fieldsReality is made by particles and fields(incldue gravity), that can only be localized with (incldue gravity), that can only be localized with respect to each other. respect to each other.

Spacetime emergence of solution of theorySpacetime emergence of solution of theory

Active Diffeormophism invariantActive Diffeormophism invariant

• Einstein field equation ： A theory on space, time and Matter

1

2R Rg kT

Geometry of Geometry of space time space time

MatterMatter

QuantizationQuantization

Quantum GravityQuantum Gravity ？？

Spacetime is curved Spacetime is curved due to the gravitational due to the gravitational action of matter action of matter

QM,QED,QCD…QM,QED,QCD…

QuantizationQuantization

(1)(1)Black holesBlack holes entropy entropy

(2)(2)CosmologyCosmology:: Big Big BBangang

SingularitySingularity

Puzzles in general relativityPuzzles in general relativity

G

AS

4

Causal Dynamical Triangularization (Lorentzian Causal Dynamical Triangularization (Lorentzian signature)signature)

Important consequences of CDT:Important consequences of CDT:

(1)(1)Baby Universe eliminatedBaby Universe eliminated(2)(2)Global Times with respect to foliationGlobal Times with respect to foliation(3)(3) No topology changeNo topology change(4)(4) New universality class then EuclideanNew universality class then Euclidean

Four-dimensional Picasso.Four-dimensional Picasso. Connecting many triangular bits of flat space-time to make curvy Connecting many triangular bits of flat space-time to make curvy space-times demonstrates how our four-dimensional world can arise space-times demonstrates how our four-dimensional world can arise from the fuzzy "quantum foam" that may exist at the tiniest scales. from the fuzzy "quantum foam" that may exist at the tiniest scales. (Image from researchers who used "non-causal" triangles.)(Image from researchers who used "non-causal" triangles.)

Causality implies four-dimensionality!Causality implies four-dimensionality!

Hausdorff dimension: V(r) ~ rV(r) ~ rddHH

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 completelydoes not degenerate completelyCausality require that each spacetime appearing in the sum Causality require that each spacetime appearing in the sum over geometry should have a specific form, namely object over geometry should have a specific form, namely object that can obtain by evolving a purely spatial geometry in time that can obtain by evolving a purely spatial geometry in time

Volume-Volume Correlator:Volume-Volume Correlator:

Another evidence for spacetime dimension 4: Another evidence for spacetime dimension 4: Spectral dimensionSpectral dimension

Effective mini-superspace and Effective mini-superspace and Wave Function of Universe (Bounce Solution):Wave Function of Universe (Bounce Solution):Tunneling of from nothing(zero scale factor a)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 GravityNonperturbative Quantum Einstein Gravity

The phase portrait of Einstein Hilbert trunctionThe phase portrait of Einstein Hilbert trunction

Now, NGFP predicts:Now, NGFP predicts:4 dimension at low energies 4 dimension at low energies 2 at high energies2 at high energies

Spectral DimensionSpectral 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.