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Todays Paper: Modified Gravity Black Holes and Their Observable Shadows by John Mo ff at[arXiv:1502.01677] Elias Roland Most 6. June 2015 Elias Roland Most Journal Club in High Energy Physics 6. June 2015 1 / 23 Galaxy Rotation Curves

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Today’s Paper: Modified Gravity Black Holes and Their Observable Shadows by John Moffat[arXiv:1502.01677]

Elias Roland Most

  • 6. June 2015

Elias Roland Most Journal Club in High Energy Physics

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Galaxy Rotation Curves

Quick calculation µv2 r = G µM r2 = ) v = r GM r Most people assume this can be explained by a dark matter halo.

Figure : M33 rotation curve [wikipedia:Galaxy Rotation Curve]

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But there are others..

Modified Newtonian Dynamics

What if for very small accelerations Newtons second law was different, F = m⇠ ✓ a a0 ◆ a where ⇠ ✓ a a0 ◆ = 1 1 + a0

a

so that for small accelerations F = m a

a0 a?

Repeating the previous calculation = ) v = (GMa0)

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Modified Gravity

An Overview

Tries to explain galaxy rotation curves without dark matter. Modifies Newton’s constant G = GN (1 + ↵) Introduces scalar fields to mimic the running of couplings normally induced by an effective theory of quantum gravity(e.g [arXiv:1202.2274]) Also introduces a vector field

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Modified Gravity

S = SGrav + S + SS + SM, where SGrav = 1 16⇡ Z d4xpg  1 G (R + 2Λ)

  • ,

S = Z d4xpg  ! ✓1 4Bµ⌫Bµ⌫ + 1 2µ2µµ + V () ◆ , and SS = Z d4xpg  1 G 3 ✓1 2gµ⌫rµGr⌫G V (G) ◆ + 1 G ✓1 2gµ⌫rµ!r⌫! V (!) ◆ + 1 µ2G ✓1 2gµ⌫rµµr⌫µ V (µ) ◆ .

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Two EoMs

Gµ⌫ gµ⌫Λ + Qµ⌫ = 8⇡GTµ⌫, where Gµ⌫ = Rµ⌫ 1

2gµ⌫R and

Qµ⌫ = G(r↵r↵Θgµ⌫ rµr⌫Θ), where Θ(x) = 1/G(x). r⌫Bµ⌫ + @V () @µ + 1 !r⌫!Bµ⌫ = 1 !Jµ.

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Geodesic equations

The test particle action is given by STP = m Z d⌧ Z d⌧!µ dxµ d⌧ . Then the geodesic equation takes the following form m ✓d2xµ d⌧ 2 + Γµ

↵

dx↵ d⌧ dx d⌧ ◆ = f µ, where f µ = !Bµ⌫ dx⌫ d⌧ + rµ! ✓ ↵ dx↵ d⌧ ◆ r↵! ✓ µ dx↵ d⌧ ◆ . Obtaining equations of motions from this is tedious!

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Massive vector fields in QFT

Detail may be found in [A.Zee: Quantum Field Theory in a Nutshell]

Start from the action S = Z d4x {µDµ⌫⌫ + µJµ} , where Dµ⌫ =

  • @2 + m2

gµ⌫ @µ@⌫. The partition function Z = R Dµ eiS is just a Gaussian integral that can easily be solved to W (J) = i log Z = 1 2 Z d4k (2⇡)4 Jµ⇤ (k) gµ⌫ + kµk⌫/m2 k2 m2 + i" J⌫ (k) . Using that J is conserved, i.e. @µJµ = 0 W (J) = 1 2 Z d4k (2⇡)4 Jµ⇤ (k) 1 k2 m2 + i"Jµ (k)

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Massive vector fields in QFT

What is the force between two massive vector field point particles?

Let J = J1 + J2, where Ji (x) = (3) (x xi). Neglecting self-interactions JiJi and using (x) = R dkeikx, W (J) = ✓Z dx0 ◆ Z d3k (2⇡)3 ei~

k·(~ x1~ x2)

~ k2 + m2 , = T emr 4⇡r

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Massive vector fields in QFT

Using that Z = ⌦ eiHT↵ we can identify E = i log Z/T = W /T = emr 4⇡r . So the potential between two massive vector particles is of Yukawa type and leads to a repulsion. Note that this only applies in flat spacetime.

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Weak field limit

To make predictions about galaxy rotation curves we need to consider the weak field limit gµ⌫ = ⌘µ⌫ + hµ⌫. Then we would also have to perturb G, µ, µ and things would get messy. Instead of doing this we can instead use our previous insights to guess Φeff (~ x) = GN (1 + ↵) Z d3x0 ⇢ (~ x0) |~ x ~ x0| + GN↵ Z d3x0⇢ ~ x0 eµ|~

x~ x0|

|~ x ~ x0| . Note that the prefactor of the second term has been chosen to cancel the the corrections at small distances, viz. emr ' 1, so that compatibility with Newtonian gravity is maintained. = ) This is indeed the result we would have obtained from the EOMs.

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Galaxy rotation curves revisited

20 40 60 80 100 120 140 160 180 5 10 15 20 25 30

R(kpc) V(km/s) NGC3726

20 40 60 80 100 120 140 5 10 15 20 25 30

R(kpc) V(km/s) NGC3769

20 40 60 80 100 120 140 160 180 1 2 3 4 5 6 7 8 9 10

R(kpc) V(km/s) NGC3877

25 50 75 100 125 150 175 200 2.5 5 7.5 10 12.5 15 17.5 20

R(kpc) V(km/s) NGC3893

20 40 60 80 100 120 140 2 4 6 8 10 12 14

R(kpc) V(km/s) NGC3917

20 40 60 80 100 120 140 160 180 1 2 3 4 5 6 7 8

R(kpc) V(km/s) NGC3949

50 100 150 200 250 2 4 6 8 10 12 14 16 18

R(kpc) V(km/s) NGC3953

20 40 60 80 100 120 140 1 2 3 4 5 6 7 8 9

R(kpc) V(km/s) NGC3972

20 40 60 80 100 120 140 2 4 6 8 10

R(kpc) V(km/s) NGC4010

Figure 2. The best fit to the rotation velocity curves of the Ursa-Major sample. We fix α = 8.89 and µ = 0.042 kpc−1 from the fits to the THINGS catalogue. We take the stellar mass-to-light ratio Υ as the free degree of freedom.

Figure : [arXiv:1306.6383]

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Galaxy rotation curves revisited

These fits fix the values of ↵ and µ to ↵ = 8.89 µ = 0.042kpc1 As distances in the solar system are of the order of light minutes, these values are compatible with the Yukawa part of the gravitational potential being absent in solar system based experiments.

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Black hole solutions[arXiv:1412.5424]

Assumptions: G = GN (1 + ↵) is constant ! = 1 µ is constant In this case the Einstein eq. reduces to Rµ⌫ = 8⇡GTµ⌫, where Tµ⌫ = 1

4⇡

  • B↵

µ B⌫↵ 1 4gµ⌫B↵B↵

  • Note that the Mass term / µ2 has been neglected.

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Reissner-Nordstr¨

  • m solution

This is nearly the same as for a charged black hole in standard GR, ds2 = ✓ 1 2GM r + GQ2 r2 ◆ dt2 ✓ 1 2GM r + GQ2 r2 ◆1 dr2 r2dΩ2. However, the charge is related to the mass by the following postulate Q = p ↵GNM.

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Reissner-Nordstr¨

  • m

The horizons are at r± = GNM h 1 + ↵ ± p 1 + ↵ i

  • 2

4 6 8 10 α 1 2 3 4 5 6 7

r rS

r+ r-

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Kerr solution

As in GR we also have a Kerr-Newman type solution ds2 = ∆ ⇢2 (dt a sin2 ✓d)2 sin2 ✓ ⇢2 [(r2 + a2)d adt]2 ⇢2 ∆ dr2 ⇢2d✓2, where ∆ = r2 2GMr + a2 + ↵GNGM2, ⇢2 = r2 + a2 cos2 ✓. horizons r± = GN(1 + ↵)M  1 ± s 1 a2 G 2

N(1 + ↵)2M2

↵ 1 + ↵

  • ergosphere

rE = GN(1 + ↵)M  1 + s 1 a2 cos2 ✓ G 2

N(1 + ↵)2M2

↵ 1 + ↵

  • .

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Kerr solutions

Horizons

  • 6
  • 4
  • 2

2 4 6 x rS

  • 6
  • 4
  • 2

2 4 6 y rS

rE r+ r- rE(GR) Elias Roland Most Journal Club in High Energy Physics

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Photons in Reissner-Nordstr¨

  • m

rph = 3 2GN(1 + ↵)M ✓ 1 + s 1 8↵ 9(1 + ↵) ◆ .

  • 10
  • 5

5 10 x rS

  • 10
  • 5

5 10 y rS

rPh rPh(GR) Elias Roland Most Journal Club in High Energy Physics

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Black hole shadow

Figure : arXiv:1502.01677

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Black hole shadow

The black hole shadow is given by the set of all closed photon orbits. We have x = r∆ + r↵G 2

N(1 + ↵)M2 GN(1 + ↵)M(r2 a2)

a[r GN(1 + ↵)M] sin ✓ , y = ⇢ 4r2∆ [r GN(1 + ↵)M]2 (x + a sin ✓)2 1/2 , where ∆ = r2 2GN(1 + ↵)Mr + a2 + ↵G 2

N(1 + ↵)M2.

For a = 0 it is given as rshad ⌘ p x2 + y2 = h 3(1 + ↵) ± p (9 + ↵)(1 + ↵) i2 ⇢ 4 h (1 + ↵) ± p (9 + ↵)(1 + ↵) i2 16(1 + ↵) 1/2 GNM.

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Figure : (a) GN = 1, M = 1, a = 0, ↵ = 0. (b) GN = 1, M = 1, a = 0, ↵ = 3. (c)GN = 1, M = 1, a = 0, ↵ = 9. (d) GN = 1, M = 1, a = 0.16, ↵ = 0 and ✓ = 63 . (e) GN = 1, M = 1, a = 0.95, ↵ = 0 and ✓ = 63 . (f) GN = 1, M = 1, a = 0.16, ↵ = 3 and ✓ = 63 . (g) GN = 1, M = 1, a = 0.16, ↵ = 9 and ✓ = 63 . (h) GN = 1, M = 1, a = 0.95, ↵ = 3 and ✓ = 63 . (i) GN = 1, M = 1, a = 0.95, ↵ = 9 and ✓ = 63 , [arXiv : 1502.01677].

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Conclusion

This implies the following important conclusion: rshad ⇠ (2.59 + 2↵)rs So for ↵ obtained from galaxy rotation curves rshad ⇠ 10rshadGR. This should be observable for Sagittarius A*.

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