modifications of gravity vs. dark matter/energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf ·...
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![Page 1: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/1.jpg)
Modifications of Gravity vs. Dark Matter/Energy
Daniel Grumiller
Massachusetts Institute of TechnologyMarie-Curie Fellowship MC-OIF 021421
Finnish-Japanese Workshop on Particle CosmologyHelsinki, March 2007
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 2: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/2.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 3: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/3.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 4: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/4.jpg)
Theoretical problems
“Standard folklore”
GR = beautiful and experimentally establishedonly “a few details” are missing
No comprehensive theory of quantum gravity
Several experimental “anomalies”
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 5: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/5.jpg)
Theoretical problems
“Standard folklore”
GR = beautiful and experimentally establishedonly “a few details” are missing
No comprehensive theory of quantum gravity
Several experimental “anomalies”
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 6: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/6.jpg)
Theoretical problems
“Standard folklore”
GR = beautiful and experimentally establishedonly “a few details” are missing
No comprehensive theory of quantum gravity
Several experimental “anomalies”
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 7: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/7.jpg)
Experimental anomalies
Dark Energy
Dark Matter
Pioneer anomaly
other anomalies?
Note intriguing coincidences:
aΛ︸︷︷︸Dark Energy
≈ aMOND︸ ︷︷ ︸Dark Matter
≈ aP︸︷︷︸Pioneer anomaly
≈ aH︸︷︷︸Hubble
≈ 10−9−10−10m/s2
in natural units: a ≈ 10−61 − 10−62
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 8: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/8.jpg)
Experimental anomalies
Dark Energy
Dark Matter
Pioneer anomaly
other anomalies?
Note intriguing coincidences:
aΛ︸︷︷︸Dark Energy
≈ aMOND︸ ︷︷ ︸Dark Matter
≈ aP︸︷︷︸Pioneer anomaly
≈ aH︸︷︷︸Hubble
≈ 10−9−10−10m/s2
in natural units: a ≈ 10−61 − 10−62
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 9: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/9.jpg)
Experimental anomalies
Dark Energy
Dark Matter
Pioneer anomaly
other anomalies?
Note intriguing coincidences:
aΛ︸︷︷︸Dark Energy
≈ aMOND︸ ︷︷ ︸Dark Matter
≈ aP︸︷︷︸Pioneer anomaly
≈ aH︸︷︷︸Hubble
≈ 10−9−10−10m/s2
in natural units: a ≈ 10−61 − 10−62
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 10: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/10.jpg)
Experimental anomalies
Dark Energy
Dark Matter
Pioneer anomaly
other anomalies?
Note intriguing coincidences:
aΛ︸︷︷︸Dark Energy
≈ aMOND︸ ︷︷ ︸Dark Matter
≈ aP︸︷︷︸Pioneer anomaly
≈ aH︸︷︷︸Hubble
≈ 10−9−10−10m/s2
in natural units: a ≈ 10−61 − 10−62
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 11: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/11.jpg)
Experimental anomalies
Dark Energy
Dark Matter
Pioneer anomaly
other anomalies?
Note intriguing coincidences:
aΛ︸︷︷︸Dark Energy
≈ aMOND︸ ︷︷ ︸Dark Matter
≈ aP︸︷︷︸Pioneer anomaly
≈ aH︸︷︷︸Hubble
≈ 10−9−10−10m/s2
in natural units: a ≈ 10−61 − 10−62
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 12: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/12.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 13: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/13.jpg)
Some experimental data of galactic rotation curves
Rotation curve of our Galaxy
Inconsistent with Newton!
(or Kepler...)
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 14: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/14.jpg)
Sketch of experimental dataA typical non-Keplerian galactic rotation curve
c©Wikipedia
Curve A: Newtonian prediction
Curve B: Observed velocity profile
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 15: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/15.jpg)
Newtonian calculation
v2︸︷︷︸kinetic
∝∫
ρdVr︸ ︷︷ ︸
potential
∝ Mr
Regime with ρ = const.: v ∝ r
Regime with M = const.: v ∝ 1/√
r
ConclusionFlat rotation curves not described well by Newton
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 16: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/16.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 17: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/17.jpg)
Saving Newton
Postulate existence of Dark Matter
Fit Dark Matter density as to “explain” rotation curves
Note: other hints for Dark Matter, e.g. gravitational lensing!
Exciting prospect for near future
Dark Matter might be discovered next year at LHC!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 18: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/18.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 19: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/19.jpg)
Example: MONDModified Newton Dynamics
~F = m~a · µ(a/aMOND)
with
µ(x) → 1 for x � 1
µ(x) → x for x � 1
critical acceleration: aMOND ≈ 10−10m/s2
Phenomenologically rather successful, but no deepertheoretical understanding
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 20: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/20.jpg)
Main issues
Two important questions:
Do we have the correct theory of gravity?
Are we applying it correctly?
Possible answers:
Regarding 1: e.g. MOND, IR modifications of GR,Yukawa-type corrections, ...
Regarding 2: Newtonian limit justified?
Consider the second point
Study appropriate General Relativistic (exact) solutions
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 21: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/21.jpg)
Main issues
Two important questions:
Do we have the correct theory of gravity?
Are we applying it correctly?
Possible answers:
Regarding 1: e.g. MOND, IR modifications of GR,Yukawa-type corrections, ...
Regarding 2: Newtonian limit justified?
Consider the second point
Study appropriate General Relativistic (exact) solutions
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 22: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/22.jpg)
Main issues
Two important questions:
Do we have the correct theory of gravity?
Are we applying it correctly?
Possible answers:
Regarding 1: e.g. MOND, IR modifications of GR,Yukawa-type corrections, ...
Regarding 2: Newtonian limit justified?
Consider the second point
Study appropriate General Relativistic (exact) solutions
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 23: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/23.jpg)
Main issues
Two important questions:
Do we have the correct theory of gravity?
Are we applying it correctly?
Possible answers:
Regarding 1: e.g. MOND, IR modifications of GR,Yukawa-type corrections, ...
Regarding 2: Newtonian limit justified?
Consider the second point
Study appropriate General Relativistic (exact) solutions
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 24: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/24.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 25: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/25.jpg)
Validity of Newtonian approximation
Locally: can use Newton almost everywhere in galaxy(except for central galactic Black Hole region and near jetaxis)
But: GR is a non-linear theory!
Not granted that Newton is applicable globally
Thus, use GR instead of Newton!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 26: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/26.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 27: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/27.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 28: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/28.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 29: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/29.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 30: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/30.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 31: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/31.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 32: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/32.jpg)
The Cooperstock-Tieu attempt
Ansatz:
Axial symmetry
Reflection symmetry
Stationarity
Pressurless perfect fluid sources
Corotating perfect fluid
Weak field limit
Claim of Cooperstock-Tieu:
GR differs essentially from NewtonHowever, technically incorrect!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 33: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/33.jpg)
Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 34: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/34.jpg)
AssumptionsH. Balasin and D. Grumiller, astro-ph/0602519
Drop weak field limit, but keep rest:
Axial symmetry → spacelike Killing
Reflection symmetry (around galactic plane)
Stationarity → timelike Killing ξa = (∂t)a
Pressurless perfect fluid sources → T ab = ρuaub
Corotating perfect fluid → ua = (u0(r , z), 0, 0, 0)
Consequently, line element can be brought into adapted form:
ds2 = −(dt − Ndφ)2 + r2dφ2 + exp (ν)(dr2 + dz2)
functions ρ, N, ν depend solely on r , z
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 35: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/35.jpg)
Einstein equations
rνz + Nr Nz = 0 , (1)
2rνr + N2r − N2
z = 0 , (2)
νrr + νzz +1
2r2
(N2
r + N2z
)= 0 , (3)
Nrr −1r
Nr + Nzz = 0 , (4)
1r2
(N2
r + N2z
)= κρeν . (5)
Important notes:
(4) is a linear PDE for N!
For known N (1)-(3) yield ν (integration constant!)
For known N, ν (5) establishes ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 36: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/36.jpg)
Einstein equations
rνz + Nr Nz = 0 , (1)
2rνr + N2r − N2
z = 0 , (2)
νrr + νzz +1
2r2
(N2
r + N2z
)= 0 , (3)
Nrr −1r
Nr + Nzz = 0 , (4)
1r2
(N2
r + N2z
)= κρeν . (5)
Important notes:
(4) is a linear PDE for N!
For known N (1)-(3) yield ν (integration constant!)
For known N, ν (5) establishes ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 37: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/37.jpg)
Einstein equations
rνz + Nr Nz = 0 , (1)
2rνr + N2r − N2
z = 0 , (2)
νrr + νzz +1
2r2
(N2
r + N2z
)= 0 , (3)
Nrr −1r
Nr + Nzz = 0 , (4)
1r2
(N2
r + N2z
)= κρeν . (5)
Important notes:
(4) is a linear PDE for N!
For known N (1)-(3) yield ν (integration constant!)
For known N, ν (5) establishes ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 38: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/38.jpg)
Einstein equations
rνz + Nr Nz = 0 , (1)
2rνr + N2r − N2
z = 0 , (2)
νrr + νzz +1
2r2
(N2
r + N2z
)= 0 , (3)
Nrr −1r
Nr + Nzz = 0 , (4)
1r2
(N2
r + N2z
)= κρeν . (5)
Important notes:
(4) is a linear PDE for N!
For known N (1)-(3) yield ν (integration constant!)
For known N, ν (5) establishes ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Inverse problemPredicting mass density from velocity profile
Physical meaning of N:
V (r , z) =N(r , z)
r
V is 3-velocity as seen by asymptotic observer at rest withrespect to center of galaxy“Inverse problem”:
Take V as experimental input (rotation curve)
Calculate mass density ρ
Compare ρ with experimental data
Compare with Newtonian prediction for ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Inverse problemPredicting mass density from velocity profile
Physical meaning of N:
V (r , z) =N(r , z)
r
V is 3-velocity as seen by asymptotic observer at rest withrespect to center of galaxy“Inverse problem”:
Take V as experimental input (rotation curve)
Calculate mass density ρ
Compare ρ with experimental data
Compare with Newtonian prediction for ρ
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Boundary conditions and asymptotics
Solution mathematically trivial (separation Ansatz)Physical input:
Modes bounded for |z| → ∞Modes bounded for r →∞No exotic δ-sources in galactic plane!
General solution for r > 0:
N(r , z) = r2
∞∫0
dxC(x)∑±
((z ± x)2 + r2)−3/2
Remaining task:
Choose spectral density C(x) appropriately!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Boundary conditions and asymptotics
Solution mathematically trivial (separation Ansatz)Physical input:
Modes bounded for |z| → ∞Modes bounded for r →∞No exotic δ-sources in galactic plane!
General solution for r > 0:
N(r , z) = r2
∞∫0
dxC(x)∑±
((z ± x)2 + r2)−3/2
Remaining task:
Choose spectral density C(x) appropriately!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Outline
1 GravityProblems with General Relativity
2 Galactic rotation curvesStatement of the problemDark matter?Alternatives
3 General relativistic descriptionMotivationAxially symmetric stationary solutionsToy model for a galaxy
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Choice of spectral density
Take simple 3-parameter family of C(x):
Parameter V0: flat region velocity (about 200km/s)
Parameter r0: bulge radius (about 1kpc)
Parameter R: choose about 100kpc
With these choices V (r , 0) looks as follows:
5 10 15 20
0.2
0.4
0.6
0.8
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Asymptotic non-flatnessThe necessity of jets
Crucial observation:
Solutions (necessarily) singular close to axis r = 0 for |z| ≥ r0
0 200 400 600 800 1000-1000
-500
0
500
1000
r
z
0 2 4 6 8 10
-4
-2
0
2
4
r
z
Cut out this region and paste there a different solution of GRSolutions are not asymptotically flat!
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Prediction for mass density
Mass density
ρ = β︸︷︷︸int. const.
N2r + N2
z
r2
on large scales:
0200
400600
8001000-1000
-500
0
500
1000
02.5·10-95·10-9
7.5·10-91·10-8
0200
400600
8001000
Integrating ρ over “white” region yields ≈ 1011 solar masses
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Comparison with Newton
For any given velocity profile V Newton predicts:
ρNewton∝V 2 + 2rVV ′
r2
Ratio of GR/Newton:
ρ
ρNewton= β̃
(1 +
r2(V ′)2
V 2 + 2rVV ′
)Assume a spectral density which yields a linear regime V ∝ rand a flat regime V = const.
Important remaining task:
Have to fix free constant β̃!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Comparison with Newton
For any given velocity profile V Newton predicts:
ρNewton∝V 2 + 2rVV ′
r2
Ratio of GR/Newton:
ρ
ρNewton= β̃
(1 +
r2(V ′)2
V 2 + 2rVV ′
)Assume a spectral density which yields a linear regime V ∝ rand a flat regime V = const.
Important remaining task:
Have to fix free constant β̃!
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Difference to Newton
Fix β̃ such that
Newton and GR coincide in the linear regime V ∝ r
This yields β̃ = 3/4.
Calculate the ratio in the flat regime V = const.
ρ
ρNewton
∣∣∣∣flat
= β̃ =34
Conclusion
Newton predicts 133% of the mass density as compared to GR
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Difference to Newton
Fix β̃ such that
Newton and GR coincide in the linear regime V ∝ r
This yields β̃ = 3/4.
Calculate the ratio in the flat regime V = const.
ρ
ρNewton
∣∣∣∣flat
= β̃ =34
Conclusion
Newton predicts 133% of the mass density as compared to GR
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
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Difference to Newton
Fix β̃ such that
Newton and GR coincide in the linear regime V ∝ r
This yields β̃ = 3/4.
Calculate the ratio in the flat regime V = const.
ρ
ρNewton
∣∣∣∣flat
= β̃ =34
Conclusion
Newton predicts 133% of the mass density as compared to GR
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 52: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/52.jpg)
Summary
Problem of flat galactic rotation curves
May be solved by Dark Matter
GR attempts to avoid Dark Matter have failed
GR predicts slightly less Dark Matter than Newton
Outlook:Improve the toy modelPerform patchingUnderstand better role of jets and central galactic blackholes for global galactic dynamics
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 53: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/53.jpg)
Summary
Problem of flat galactic rotation curves
May be solved by Dark Matter
GR attempts to avoid Dark Matter have failed
GR predicts slightly less Dark Matter than Newton
Outlook:Improve the toy modelPerform patchingUnderstand better role of jets and central galactic blackholes for global galactic dynamics
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 54: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/54.jpg)
Summary
Problem of flat galactic rotation curves
May be solved by Dark Matter
GR attempts to avoid Dark Matter have failed
GR predicts slightly less Dark Matter than Newton
Outlook:Improve the toy modelPerform patchingUnderstand better role of jets and central galactic blackholes for global galactic dynamics
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 55: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/55.jpg)
Summary
Problem of flat galactic rotation curves
May be solved by Dark Matter
GR attempts to avoid Dark Matter have failed
GR predicts slightly less Dark Matter than Newton
Outlook:Improve the toy modelPerform patchingUnderstand better role of jets and central galactic blackholes for global galactic dynamics
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 56: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/56.jpg)
Summary
Problem of flat galactic rotation curves
May be solved by Dark Matter
GR attempts to avoid Dark Matter have failed
GR predicts slightly less Dark Matter than Newton
Outlook:Improve the toy modelPerform patchingUnderstand better role of jets and central galactic blackholes for global galactic dynamics
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy
![Page 57: Modifications of Gravity vs. Dark Matter/Energyquark.itp.tuwien.ac.at/~grumil/pdf/finnland07.pdf · 2007-03-28 · Modifications of Gravity vs. Dark Matter/Energy Daniel Grumiller](https://reader035.vdocuments.net/reader035/viewer/2022070914/5fb4fd2aa07f577e8c0c8721/html5/thumbnails/57.jpg)
Literature I
C. Lammerzahl, O. Preuss and H. Dittus, “Is the physicswithin the Solar system really understood?,”gr-qc/0604052 .
F. I. Cooperstock and S. Tieu, astro-ph/0507619 .
M. Korzynski, astro-ph/0508377 ; D. Vogt andP. S. Letelier, astro-ph/0510750 .
H. Balasin and D. Grumiller, astro-ph/0602519 .
Daniel Grumiller Modifications of Gravity vs. Dark Matter/Energy