**Author details**

*∂*

Here,

*Progress in Relativity*

O*<sup>γ</sup>*

so we see that

where the quantity O*<sup>γ</sup>*

general, nonlinear in *n<sup>μ</sup>*. We therefore have

contribute in the limit of the an Abelian gauge. Returning to the variation of L*<sup>f</sup>* , we see that

the fields given the source currents:

*j*

*<sup>μ</sup>* ¼ �*<sup>i</sup>* <sup>ϵ</sup>

where, from (106),

**6. Summary**

**156**

*<sup>δ</sup>*L*<sup>f</sup>* ¼ � <sup>1</sup>

þ *f*

where we have taken into account the fact that *nμ; n<sup>μ</sup>*

and integrated by parts the derivatives of *δn*. We then obtain

*<sup>δ</sup>*L*<sup>f</sup>* ¼ �*∂<sup>ν</sup>*

*∂ν f μν* ¼ *j*

<sup>2</sup>*<sup>M</sup> <sup>ψ</sup>* <sup>∗</sup> *<sup>∂</sup><sup>μ</sup>* � *<sup>i</sup>*ϵ*n<sup>μ</sup>*

¼ �*∂<sup>ν</sup>*

*<sup>∂</sup>n<sup>γ</sup> <sup>n</sup>*0*<sup>μ</sup>; <sup>n</sup>*0*<sup>ν</sup>* ½ �¼ <sup>2</sup>*i*<sup>ð</sup> *δν*

*∂v<sup>μ</sup> <sup>∂</sup>n<sup>γ</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>0</sup> *<sup>γ</sup>* ð Þþ *<sup>n</sup>* � *<sup>b</sup> <sup>n</sup><sup>ν</sup>*

<sup>þ</sup> *<sup>k</sup><sup>ν</sup> <sup>∂</sup>v<sup>μ</sup>*

*μ <sup>γ</sup>* þ *ω*<sup>00</sup> *λ μ <sup>n</sup><sup>λ</sup> <sup>∂</sup>k<sup>σ</sup> <sup>∂</sup>n<sup>γ</sup> <sup>x</sup><sup>σ</sup>,*

*<sup>∂</sup>n<sup>γ</sup> <sup>n</sup>*0*<sup>μ</sup>; <sup>n</sup>*0*<sup>ν</sup>* ½ �� <sup>O</sup>*<sup>γ</sup>*

*<sup>δ</sup><sup>n</sup> <sup>n</sup>*0*μ; <sup>n</sup>*0*<sup>ν</sup>* ½ �¼ <sup>O</sup>*<sup>γ</sup>*

<sup>4</sup> <sup>ð</sup> *<sup>∂</sup>μδn<sup>ν</sup>* � *<sup>∂</sup><sup>ν</sup>*

In the limit that *ω* ! 0, its derivative and higher derivatives which appear in

*μν* may not vanish (somewhat analogous to the case in gravitational theory when the connection form vanishes, but the curvature does not), so that this term can

*<sup>δ</sup>n<sup>μ</sup>* <sup>þ</sup> *<sup>i</sup>*ϵ*<sup>δ</sup> <sup>n</sup>μ; <sup>n</sup><sup>μ</sup>* ½ � *<sup>f</sup> μν*

*μν <sup>∂</sup>μδn<sup>ν</sup>* � *<sup>∂</sup>νδn<sup>μ</sup>* <sup>þ</sup> *<sup>i</sup>*ϵ*<sup>δ</sup> <sup>n</sup>μ; <sup>n</sup><sup>μ</sup>* <sup>Þ</sup>

*<sup>f</sup> μνδn<sup>μ</sup>* <sup>þ</sup> <sup>2</sup>*if μν<sup>δ</sup> <sup>n</sup>μ; <sup>n</sup><sup>ν</sup>* ½ �*,*

Since the coefficient of *δn<sup>μ</sup>* must vanish, we obtain the Yang-Mills equations for

which is nonlinear in the fields *n<sup>μ</sup>*, as we have seen, even in the Abelian limit,

In this chapter, we have shown that the formulation of MOND theory by Bekenstein and Milgrom [5–10] can have a systematic origin within the framework of the embedding of the SHP [1] theory into general relativity [20]. The SHP

*<sup>ψ</sup>* � *<sup>∂</sup><sup>μ</sup>* <sup>þ</sup> *<sup>i</sup>*ϵ*n<sup>μ</sup>*

*∂*

*v<sup>μ</sup>*

*<sup>∂</sup>n<sup>γ</sup>* � ð ÞÞ *<sup>μ</sup>* \$ *<sup>ν</sup> :*

*μνδn<sup>γ</sup>* depends on the first and second derivatives of *ω<sup>μ</sup>*

*bγ*

*μν,* (114)

*μνδn<sup>γ</sup>* (115)

is a c-number function

*<sup>f</sup> μνδn<sup>μ</sup>* <sup>þ</sup> <sup>2</sup>*i*ϵ*<sup>f</sup> λσ*O*λσμδn<sup>μ</sup>* (117)

*<sup>μ</sup>* � <sup>2</sup>*i*ϵ*<sup>f</sup> λσ*O*λσμ,* (118)

*ψ* <sup>∗</sup> *ψ :* (119)

(113)

*<sup>λ</sup>* , in

(116)

Lawrence P. Horwitz1,2,3

1 School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel

2 Department of Physics, Bar Ilan University, Ramat Gan, Israel

3 Department of Physics, Ariel University, Ariel, Israel

\*Address all correspondence to: larry@tauex.tau.ac.il

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
