Download - Emergent Gravity according to Verlinde
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EMERGENT GRAVITYErik Verlinde’s Proposal
Stephen Perrenod, Ph.D.
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Initial Concept
• Erik Verlinde, in 2011, published a paper on Emergent Gravity
• “On the Origin of Gravity and the Laws of Newton”
• Motivations - connections to string theory, need for quantum description of gravity and the uncertainty around dark matter
• He derived Newton’s laws in an emergent gravity scenario, but assumed an unrealistic Anti-deSitter space
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Quantum Entanglement
• Quantum entanglement is when pairs (or groups) of particles have their quantum states coupled
• The quantum states of the two particles remain coupled as they are separated and observing a property of one is tantamount to observing both
• This has been observed in the laboratory over long distances of ~ 100 kilometers
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Emergence• Space is defined by the connectivity between
quantum entangled particles
• Gravity is an emergent, statistical feature of the system
• Just as thermodynamics is an emergent feature of the particles of air in the atmosphere
• The entropy (and information) content of space is due to the excitations of the vacuum state that manifest dark energy
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de Sitter and Anti-de Sitter
• de Sitter space has a positive Cosmological Constant (Λ > 0), as does our universe
• Anti-de Sitter space (AdS) with negative Λ has been popular with string theorists
• because easier to solve and they can show a correspondence between AdS and conformal field theory (CFT)
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Anti-de Sitter
• In the AdS/CFT correspondence quantum entanglement of the microscopic state leads to:
• Spacetime geometry is due to the entanglement structure of the microscopic quantum state
• Entanglement is the glue for the connectivity of spacetime!
• Gravity is then an emergent phenomenon rather than a fundamental force
• Determined by the entropy on the surface surrounding the matter
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de Sitter
• Our universe is not AdS, but roughly a dark energy dominated de Sitter solution (dS)
• Verlinde has now developed a partial solution for the dS case also
• And he finds that there is a volume (bulk) contribution as well as a surface contribution
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Entropy
• Entropy is the information content reflecting the range of states, the number of microscopic degrees of freedom
• The volume contribution grows relative to the surface (area) contribution as one looks at a larger and larger volume (R3 vs. R2)
• The Hilbert space of a full system has entanglement scaling with volume, while ground states scale with area
• Newtonian dynamics / general relativity reflect the surface contribution
• There is an additional term, resulting in a stronger gravitational field than predicted by GR, when the volume contribution comes into play
• This ‘spoofs’ the existence of dark matter, but there is no DARK MATTER !
GibbsBoltzmann
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Black Holes
• The connection between gravity and thermodynamics has been around for 3 decades
• This is originally from the black hole research of Jacob Bekenstein and Stephen Hawking
• The Entropy is proportional to the black hole area divided by the Gravitational constant (G)
• The AdS/CFT correspondence in string theory derives the same formula for quantum entanglement in a vacuum
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Cosmic Horizon• The cosmic horizon is at radius ~ c/H0 where H0 is the Hubble
parameter (~ 70 kilometers/sec/Megaparsec and a parsec is 3.26 light-years), and it is around 4.1 Gigaparsecs
• So it is only at very large scales, in the outer regions of galaxies and at larger scales that departures from GR can readily be seen
• Generally these departures from standard gravitational theory are interpreted as an unseen matter contribution: Dark Matter
• But MOND (Modified Newtonian Dynamics) has been around as a phenomenological interpretation due to Milgrom since the 1980s, who argued that at very low accelerations of order a centimeter/sec/year one should modify the gravitational law
• Verlinde’s proposal is not MOND, it provides an actual theoretical basis, unlike MOND, but it does predict similar behavior
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Dark Energy
• Dark Energy (Λ) has been around since early in the history of general relativity
• It was observed through plotting the redshift-luminosity relation for distant supernovae in 1998, and subsequently in fitting to the cosmic microwave background measurements
• The CMB measurements are well fit by the simple cosmological constant model in GR with an unchanging value of Λ (equation of state pressure = - energy density)
• In the canonical ΛCDM model with cold dark matter (CDM) it represents about 69% of the universe’s mass-energy and CDM represents about 26%
• However, if Verlinde is right, then there is only ordinary matter at 5% and then Λ would be around 95% of the mass-energy!
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Cosmological Constant Λ
• The cosmological constant (dark energy for an equation of state w = -1 as observed) has units of inverse length squared
• The Hubble parameter H0 has units of inverse time
• Thus a cosmic acceleration scale in a dS universe dominated by Λ is:
• H2 / sqrt (Λ)
• The observed values are:
• Λ =1.2 x 10-56 cm-2 and H0 = 2.2 x 10-18 sec-1
• 4 x 10-8 cm / sec / sec
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Cosmic Horizon & Acceleration
• The cosmological horizon is at distance of order c/H0 (currently the ‘particle horizon’ is at about 3.2 times that distance in the standard cosmological model)
• With a cosmological acceleration scale defined by a0 = c*H0
• For H0 = 70, this works out to be a0 ~ 20 kilometers2/sec
2/parsec (in
astronomer units) or ~ 2 centimeters/sec/year (everyday units) or ~ 6 x 10
-8 cm / sec / sec
• Over 15 billion times weaker than Earth’s surface gravity !
• This is same order of magnitude as MOND which finds a characteristic acceleration of about 10
-8 cm / sec / sec as the key
parameter to use for rotation curves of galaxies and other dynamics, it also works for gravitational lensing observations
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Cosmic Acceleration Scale
• Coincidence?
• Not really, we live in a Dark Energy dominated (dS) universe
• ΩΛ ~ 0.7 if general relativity w/ dark matter, where the Ω notation is energy/mass density in units of the critical density (9 x 10-30 gm/cc or 5 protons per cubic meter). Standard ΛCDM model.
• ΩΛ ~ 0.95 if no dark matter and emergent gravity
• In a fully dS (ΩΛ = 1) universe Λc2 = 3 H02 so the two estimates are
equivalent to within sqrt(3) which is roughly their numerical difference
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String Theoretic View
• The dark energy excitations are fundamental
• Matter particles are bound states of those excitations that have sprung forth from the DE medium
• They can move freely and have much smaller entropy
• Matter creation removes energy and entropy from the underlying DE medium
• GR describes response of area law entanglement of the vacuum to matter
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Dark Energy
• Verlinde proposes that Λ and the accelerated expansion of the universe are due to the slow rate at which the emergent spacetime thermalizes
• High degeneracy and very slow dynamics means the universe is not in a ground state, therefore there should be a volume contribution to entropy
• At large distances, the volume contribution overwhelms, and since the time scale for dynamics is 1/H0 then a distance scale of c/H0 is natural
• It is also interesting that since a dS universe has a constant Λ , as well as H, then if Λ is 95% of mass-energy in our universe, we are very close to this pure dS solution
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Surface Mass Density
• When the surface mass density falls below c*H0/(8πG) things change
• The spacetime medium becomes elastic
• The effective ‘dark’ gravity is calculated resulting from an effective surface density additional contribution proportional to the square root of the baryon (ordinary matter) surface density
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Modified Acceleration
• With this new effective surface density the acceleration for dark gravity scales with the Hubble acceleration as:
• gD = sqrt (gB*a0/6) and the total acceleration is thus g = gB + gD (B baryons, D dark)
• And a0/6 is 10-8 cm/sec/sec, this approximate cutover is quite close to the MOND favored value 1.2 x 10-8
• Example, suppose gB is 10-8 cm/sec/sec, then gD is also 10-8 and the total gravity is g = gB + gD is doubled to g = 2 * 10-8 cm/sec/sec, spoofing the appearance of additional dark matter equal to ordinary matter
• But it is not DM , it is additional gravity due to the volume contribution to entropy which ordinary baryonic matter has replaced and displaced
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Glassy Dynamics, Memory Effects
• Extremely long-range correlations of the underlying microscopic degrees of freedom / entanglement
• Slow dynamics of many delocalized microscopic states whose presence and dynamics invisible at small scales
• These carry the dS entropy in a non-local way
• Glassy systems have slow relaxation and memory effects
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Galaxy Rotation Curves• Dark Matter was first proposed by
Zwicky in the 1930s to explain clusters of galaxies
• And later in the 1970s Vera Rubin and others did work on galaxy rotation curves
• In both cases there should be of order ~ 5 times more matter or ~ 5 times more gravity than seen in stars, gas
• It has also been known since the 1980s that MOND gives a good phenomenological description of rotation curves
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Tully-Fisher Relation
• Tully and Fisher observed that the luminosity of a spiral galaxy goes as the 4th power of angular velocity (or emission line width)
• Since luminosity is basically proportional to ordinary baryonic mass of a galaxy, and since with Newtonian gravity the relationship should be M ~ v2 rather than v4
it suggests
either dark matter or a deviation from standard gravity
• MOND predicts v4 for low acceleration regimes
• Verlinde is able to derive a Tully-Fisher relationship for emergent gravity by modeling the displacement of the entropy content of a dS space
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Clusters of Galaxies• It is well known that the MOND formula
falls short by a factor of 2 to 3 in clusters of galaxies
• Emergent gravity can do better, for a general mass distribution, rather than a point mass
• And one picks up a factor of 1.5 to 3.5 times MOND
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Universe
• To first order he calculates what the universal value of apparent dark matter should be
• ΩD2 = (4/3) ΩB ,
• Here B is for the baryonic matter, and for ΩB ~ 0.05 one finds ΩD ~ 0.26, these are very close to the actual measured values with the canonical ΛCDM interpretation of the Planck CMB results
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Approximations• Pure dS
• Isolated, Static region (e.g. a bound system)
• Spherical geometry
• Not clear how to explain the CMB density fluctuations and galaxy formation
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Breaking News
• Margot Brouwer et al. 2016 “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements”
• Prediction from EG, using no free parameters, is in good agreement with galaxy-galaxy lensing across a sample of surface mass density profiles of 33,613 isolated galaxies
• GR can also fit the lensing data, but has more parameters
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Summary
“The observed phenomena that are currently attributed to dark matter are the consequence of the emergent
nature of gravity and are caused by an elastic response [of the dark energy medium] due to the volume law
contribution to the entanglement entropy in our universe.” - Erik Verlinde
[my addition to statement, my underlining]
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References
• Erik Verlinde 2011 “On the Origin of Gravity and the Laws of Newton” arXiv:1001.0785
• Stephen Perrenod, 2013, 2nd edition, “Dark Matter, Dark Energy, Dark Gravity” Amazon (read Dark Matter chapter with skepticism!)
• Erik Verlinde 2016 “Emergent Gravity and the Dark Universe arXiv:1611.02269v1
• Margot Brouwer et al. 2016 “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements” arXiv:1612.03034v
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Friedmann Equation• In general relativity the relation between the critical matter and energy density and
the Hubble constant (expansion parameter) is:
• (8π/3) G ρ = H0
2
• where ρ is the total mass-energy density written in the form of a matter density
• And for a dark energy dominated universe of energy density ρde
• (8π/3) G (ρde/c2
) = H0
2
• or, G ρde = (3/8π) (c H0)2
= (3/8π) (a0)2
written in terms of the Hubble acceleration, defined to be c*H0 (since c has units cm/sec and H0 units sec-1)
• This suggests that a0 , the Hubble acceleration, is the characteristic acceleration scale for the interaction between ordinary gravity G and dark energy ρde
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Stress and Strain
• The displacement of entropy is not erased, but leaves a memory imprint, a residual stress and strain
• The medium exerts a reaction force on the matter, which is the excess or dark, gravity (normally blamed on ‘dark matter’)
• Elastic back-reaction, he calculates shear modulus ~ a0
2 / (16 πG)