On the Difference between Time and Space

Asher Yahalom

Ariel University, Ariel 40700, Israel

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Recent Developments in General Relativity

In Memory of Joseph Katz (1930-2016)

May 21-23, 2017

Beit Belgia , Edmond J. Safra Campus, Hebrew University, Jerusalem, Israel

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Content The difference between time and space The stability approach Matter included The maximal size of space time. Metric sign changes Implications Cosmology and the horizon homogeneity

problem.

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Bibliography[1] Asher Yahalom "The Geometrical Meaning of Time" [“The Linear Stability of Lorentzian Space-Time” Los-Alamos Archives - gr-qc/0602034, gr-qc/0611124] Foundations of Physics http://dx.doi.org/10.1007/s10701-008-9215-3 Volume 38, Number 6, Pages 489-497 (June 2008).

]2[ Asher Yahalom "The Gravitational Origin of theDistinction between Space and Time" InternationalJournal of Modern Physics D, Vol. 18, Issue: 14, pp.2155-2158 (2009). DOI:10.1142/S0218271809016090.

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Bibliography[3] Asher Yahalom "Gravity and the Complexity ofCoordinates in Fisher Information" International Journalof Modern Physics D, Vol. 19, No. 14 (2010) 2233–2237, © World Scientific Publishing CompanyDOI: 10.1142/S0218271810018347.

]4[ Asher Yahalom "The geometrical meaning of time -the emergence of the concept of time in the generaltheory of relativity" a chapter in a book "Advancesin Classical Field Theory", Bentham eBooks eISBN: 978-1-60805-195-3,2011. http://www.bentham.org/ebooks/9781608051953/index.htm.

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Bibliography[5] Yahalom, A. (2013). Gravity and Faster than LightParticles, Journal of Modern Physics (JMP), Vol. 4 No. 10PP. 1412-1416. DOI: 10.4236/jmp.2013.410169.[6] Asher Yahalom “On the Difference between Timeand Space” Cosmology 2014, Vol. 18. 466-483.Cosmology.com.[7] Asher Yahalom “The Geometrical Meaning of Time -Some Cosmological Implications” Proceedings of 3rd

International Conference on Mathematical Modeling inPhysical Sciences (IC-MSQUARE 2014), 28–31 August2014, Madrid, Spain. Journal of Physics: ConferenceSeries (IOP Publishing), Volume 574, 012061, 2015.doi:10.1088/1742-6596/574/1/012061.

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Bibliography[8] A. Yahalom “The Geometrical Meaning of Time inthe Presence of Matter” Journal of Physics: ConferenceSeries 633 (2015) 012030 doi:10.1088/1742-6596/633/1/012030 IOP Publishing.[9] Asher Yahalom “Preliminary Stability Analysis of aFriedman-Lemaitre-Robertson-Walker Universe”Accepted to the Proceedings of the 10th BiennialConference on Classical and Quantum RelativisticDynamics of Particles and Fields, 6 - 9 June 2016,Ljubljana, Slovenia.[10] Asher Yahalom “Gravity, Stability and CosmologicalModels” – Gravity research foundation honorablemention (2017).

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The difference between time and space

It is well known that our daily space-time isapproximately of Lorentz (Minkowski) type thatis, it possesses the metric ηµν= diag (1,-1,-1,-1).The above statement is taken as one of thecentral assumptions of the theory of specialrelativity and has been supported by numerousexperiments.

But why should it be so?

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Hermann Minkowski1864 - 1909

“The views of space and time which Iwish to lay before you have sprungfrom the soil of experimental physics,and therein lies their strength. Theyare radical. Henceforth space by itself,and time by itself, are doomed to fadeaway into mere shadows, and only akind of union of the two will preservean independent reality.”

Minkowski's address delivered at the 80th

Assembly of German Natural Scientists andPhysicians (21 September 1908)

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What does it mean a “kind ofunion”?

And if united why are space andtime different?

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One may reply, well that is theway things are, time and spaceare obviously different, whyshould we bother ourselves?

But this is not a scientificapproach!!

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For instance the mass of the electron isknown empirically to a reasonably highaccuracy but this does not explain whyelectrons have such a mass nor does itexplain why electrons exist at all.

To explain such phenomena researcherstry to construct theories such as stringtheory which hopefully will yield anexplanation.

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Thus well established empirical factsdemand theories to explain them. Toput this in other words, science is notjust a collection of well establishedfacts, rather it is a struggle to explainthose well established facts using aminimal number of assumptions.

The emphasis is on the word minimalbecause explaining well establishedempirical facts with an arbitrarynumber of assumptions is not achallenge at all.

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William of Ockham1285 - 1347

Ockham’s Razor:

“Plurality is not to be posited without necessity.”

Don’t multiply complex causes to explain things when a simple one will do

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Albert Einstein1879 - 1955

“It can scarcely be denied thatthe supreme goal of all theoryis to make the irreduciblebasic elements as simple andas few as possible withouthaving to surrender theadequate representation of asingle datum of experience.”

“On the Method of Theoretical Physics” the Herbert Spencer Lecture, Oxford, June 10,1933.

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Did Albert Einstein practiced hispreaching in the construction ofthe General Theory of Relativityor could he do better?

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Most textbooks (and also Joseph Katzlecture notes of the beginning of the1990’s) state that in the general theory ofrelativity any space-time is locally of thetype:

ηµν= diag (1,-1,-1,-1)

although it can not be presented soglobally due to the effect of matter. Thisis a part of the demands dictated by thewell known equivalence principle.

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Riemannian Geometry is replaced byPseudo Riemannian Geometry.

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In the case of Riemannian Geometry:

𝒅𝒅𝒅𝒅𝟐𝟐=𝒅𝒅𝒙𝒙𝟐𝟐+𝒅𝒅𝒚𝒚𝟐𝟐

𝒅𝒅𝒙𝒙

𝒅𝒅y

𝒅𝒅𝒅𝒅

Pythagoras' theorem

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In the case of our Pseudo Riemannianspace time something strange happens:

𝒅𝒅𝒅𝒅𝟐𝟐=𝒅𝒅𝒙𝒙𝟐𝟐 − 𝒅𝒅𝒚𝒚𝟐𝟐

𝒅𝒅𝒙𝒙

𝒅𝒅y

𝒅𝒅𝒅𝒅 Why?

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Because we know that nature obeys(locally) the rules of specialrelativity for “empty” space-time.

Wrong answer!

(Equivalent to saying that things arewhat they are because that is howthings are)

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Never the less the above principle istaken to be one of the assumptions ofgeneral relativity other assumption suchas diffeomorphism invariance, and therequirement that theory reduce toNewtonian gravity in the proper regimelead to the Einstein equations:

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Can we reduce the numberof assumptions as Einsteinrecommends?

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The stability approachAt about the same time as I was taking JosephKatz course in General relativity I was also doingmy MSc followed by a PhD under the supervisionof Joseph in the stability of stationary selfgravitating flows (Galactic Models).

It then occurred to me that the question of QuasiRiemannian geometry may be connected to thenotion of stability.

It took me more than ten years and a Sabbaticalin Cambridge to be able to formulate the problemmathematically.

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The stability approach

Let us look at the possible constant metricsavailable in the general theory of relativity whichare not equivalent to one another by a trivialtransformation, that amounts to a simple changeof coordinates.

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Since the metric is a symmetric matrixwe can diagonalize it using a unitarytransformation in which both thetransformation matrix and theeigenvalues obtained are real. Thuswithout loss of generality we can assumethat in a proper coordinate basis:

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By changing the units of the coordinates(scaling), we can always obtain:

Notice that a zero eigen-value is notpossible due to our assumption that thespace is four dimensional.All those metrics are perfectly goodsolutions of Einstein Equations for a flatspace.

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So the only real freedom is in the signsof the diagonal elements. In the case ofan Euclidian space-time all signs arepositive hence:

𝒅𝒅𝒅𝒅𝟐𝟐=𝒅𝒅𝒙𝒙𝟐𝟐+𝒅𝒅𝒚𝒚𝟐𝟐

𝒅𝒅𝒙𝒙

𝒅𝒅y

𝒅𝒅𝒅𝒅

Pythagoras' theorem

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In the case of our real space-timesomething strange happens:

𝒅𝒅𝒅𝒅𝟐𝟐=𝒅𝒅𝒙𝒙𝟐𝟐 − 𝒅𝒅𝒚𝒚𝟐𝟐

𝒅𝒅𝒙𝒙

𝒅𝒅y

𝒅𝒅𝒅𝒅

Why does nature chooses this absurd solution?

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In the lack of matter Einstein Equationsbecome:

This “almost” always true as the lefthand side is small unless for extremecases (black holes etc.).We make a small perturbations to anyflat space time in order to study itsstability:

The empty space case

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In which we assumed a proper gauge.

Now we can write the Einstein equationsin terms of this expression (keeping firstorder terms):

We define:

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We solve the above equation usingFourier decomposition:

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Which take the form:

Choosing:

We see that the only way to avoidexploding solutions is to choose :

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Choosing:

We see that the only way to avoidexploding solutions is to choose :

Hence the only stable type of metric isthe Lorentz type.

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The existence of time isnot an assumption ofgeneral relativity it is aresult of general relativity!

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Question: why should we not assume“boundary conditions” that will take care ofthe unstable perturbation even in theEuclidean case?

Answer 1: How should nature know toimpose such boundary conditions in everypoint of space-time?

Answer 2: Special boundary conditions is anadditional assumption. Additionalassumptions is something we are trying toavoid.

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Joseph Katz1930-2016

Question: Linear stableconfigurations can be nonlinearlyunstable and linear unstableconfigurations can nonlinearlystable. So is linear stabilityanalysis sufficient?

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Answer 1: Lorentz metric is stable also non-linearly. See Christodoulou, D. & Klainerman,S. (1989-1990). The global nonlinearstability of the Minkowski space, SeminaireEquations aux derivees partielles (dit"Goulaouic-Schwartz"), Exp. No. 13, p. 29.

Answer 2: Non-linear general relativity isonly significant in extreme cases (big bang,black holes etc..) this is very remote fromthe empty space scenario underinvestigation. (my thanks to Donald forpointing this out).

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Answer 3: As the question of non-linearstability of Lorentz space-time is settled allwe have to do as Joseph pointed out is toprove nonlinear instability for a single typeof perturbation, work under progress…

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Matter includedSo far we have discussed the case of stability ofthe Lorentzian metric in empty space.

The result obtained is rather philosophical, butdoes it have any physical consequences?

We expect that this result should be valid also tothe case of an almost empty space which is thegeneric situation but we cannot be sure unlesswe check.

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Let us assume a fluid energy momentum tensorof the form:

In the above p is the pressure, ρ is the densityand 𝑢𝑢𝜇𝜇 = 𝑑𝑑𝑑𝑑𝜇𝜇

𝒅𝒅𝒅𝒅, in which the interval ds is defined

as:

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We notice that 𝑇𝑇𝜇𝜇𝝂𝝂depend on metric perturbationsboth directly through the term −𝑝𝑝 𝒈𝒈𝝁𝝁𝝂𝝂 and throughthe ds term.

If we assume a uniform density then we have aFriedman-Lemaitre-Robertson-Walker metric.

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Perturbation leads to the following equations:

a,b are spatial indices.

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In the presence of matter stability analysis involves a critical wave number (in the case that pressure is negligible with respect to density) :

In terms of wavelengths we see that an upper scale for stable perturbations is:

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The maximal size of space timeThe density of the universe is estimated to be (”Universe 101: What is the Universe Made Of?”. NASA: WMAP’s Universe. Jan 24, 2014. Retrieved 17/2/2015).

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Which leads to:

This is slightly larger than the radius of theobservable universe:(Bars, I. & Terning, J. (2009). Extra Dimensions in Space and Time. Springer. pp. 27. ISBN 978-0-387-77637-8. Retrieved 2011-05-01).

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Intermediate Conclusion

Hence in ”surprising” coincidence the diameter ofthe observable universe is about the size of thelargest scale stable perturbation. At this time thesize of the universe is not known but it issuspected that above the stability scale themetric of space-time and hence physics may bequite different.

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Simple Perturbations

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http://adamdempsey 90/ygrene_krad/nohtyp/oi.buhtig.atadeht#lmth.ygrene_krad

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A Cosmological Constant

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Conditions of Stability:

Hence the Lambda-CDM model is not stable

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Some more perturbations

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Neglecting radial perturbations we obtain the stability condition:

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High density regions of space time may supportmetrics with different signatures with respect toLorentz.

Conjecture

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Metric sign changesSo far we have discussed the stability of aLorentzian Space-Time, but are there solution ofGR in the metric is not Lorentz in some part ofspace time?In other words are there metric sign changes?And if so what are the physical implications ofthis?

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A Black HoleThe Schwarzschild square interval (in terms ofspherical coordinates) is given by:

In which the Schwarzschild radius is given by:

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For r>rs we have a local metric of the type:

Which is a Lorentz type metric. For r<rs we havea local metric of the type:

Which is also a Lorentz type metric.But notice the shocking exchange of space andtime! Signs have change but not the nature ofthe metric.

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ImplicationsLet us look at the action:

And assume a general constant metric. Thisaction implies the following set of equations:

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𝝉𝝉 is a parameter along the trajectory usuallydefined as the absolute value of the interval(the proper time):

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The subluminal Lorentz case:

This is the standard case. A particle which issubluminal will always stay subluminal.

At t=0 the particle is subluminal.

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The superluminal Lorentz case:

This is a non standard case. A particle which issuperluminal will always stay superluminal.

At t=0 the particle is superluminal.

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The Euclidean case:

A particle is indifferent to whether it issuperluminal or subluminal and its velocitycan cross the speed of light from eitherdirection.

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Consider the following scenario:A particle is accelerated to a velocityclose to the velocity c in a Lorentzspace, enters into an Euclidean spaceand accelerated further in this region tovelocities above the speed c andemerge in a Lorentz space in which itwill remain above the speed c for everunless it is decelerated in an Euclideanspace again.

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The homogeneity (horizon) problem

According to an analysis given byNarlikar a proper radius for a particlehorizon of a sub luminal particle at theradiation dominated epoch was RL =2ct, taking into account temperatures ofthe early universe led him to concludethat this radius was of the order ofmagnitude of about 1 meter on presentday scales.

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In reality the cosmological micro wavebackground is homogeneous on a scaleof 1026 meters.

Notice, however, that superluminalparticles are not restricted by thevelocity of light and hence can bring avery young universe into thermalequilibrium.

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Conclusion

1. Time should not be assumed, it should bederived (the direction of time was notdiscussed but is the result of the H-theoremcombined with the low entropy conditions atthe big-bang)

2. Space-time has a finite radius, above of whichphysics is quite different.

3. A possible explanation to the homogeneity ofthe cosmic microwave background (horizonproblem) is suggested. Of course many detailsof this explanation need to be worked out.

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Conclusion4. A more popular explanation is the “inflation”

model suggested by Alan Guth, postulatingone or more scalar fields which are needed inorder to provide an explanation.

5. The original model of Guth suffered from anentropy problem (predicted too muchentropy). Later suggestions by Linde sufferedfrom the need to fine tune the parameters.

6. Moreover, it was shown that a possibleexplanation within the frame-work of standardCosmology does exist for the horizon problemif one looks closely at the metric changes ofthe Friedman-Lemaitre-Robertson-Walkermetric.

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Conclusion7. A basic flaw in common to all inflation models.

All inflation models require to postulate one ormore scalar fields which have no function,implication or purpose in nature except fortheir ad-hoc use in the inflation model. This isin sharp contradiction with the principle ofOckham's razor which demand that aminimum number of assumptions will explaina maximum number of phenomena.Postulating a physical field for everyphenomena does not serve the purpose oftheoretical science.

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William of Ockham1285 - 1347

lex parsimoniae

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Albert Einstein1879 - 1955

“It is not the result of scientificresearch that ennobles humansand enriches their nature, butthe struggle to understand whileperforming creative and openminded intellectual work.”

Mein Weltbild, 1934, 14.