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1.1 Heuristics of higher-dimensional gravity

One of the fundamental aspects of string theory is the need for extra spatial dimensions. This revives the original higher-dimensional ideas of Kaluza and Klein in the 1920s, but in a new context of quantum gravity. An important consequence of extra dimensions is that the 4-dimensional Planck scale Mp ≡ M4 is no longer the fundamental scale, which is M4+d, where d is the number of extra dimensions. This can be seen from the modification of the gravitational potential. For an Einstein–Hilbert gravitational action we have
1 ∫ ∘ -------[ ] Sgravity = ---2-- d4x ddy − (4+d)g (4+d)R − 2Λ4+d , (1 ) 2 κ4+d (4+d) (4+d) 1 (4+d) (4+d) (4+d) 2 (4+d) GAB ≡ RAB − -- R gAB = − Λ4+d gAB + κ4+d TAB, (2 ) 2
where A μ 1 d X = (x ,y ,...,y ), and 2 κ 4+d is the gravitational coupling constant,
κ2 = 8πG4+d = -8π--. (3 ) 4+d M 24++dd
The static weak field limit of the field equations leads to the 4 + d-dimensional Poisson equation, whose solution is the gravitational potential,
2 κ-4+d- V(r) ∝ r1+d . (4 )
If the length scale of the extra dimensions is L, then on scales r ≲ L, the potential is 4+d-dimensional, −(1+d) V ∼ r. By contrast, on scales large relative to L, where the extra dimensions do not contribute to variations in the potential, V behaves like a 4-dimensional potential, i.e., r ∼ L in the d extra dimensions, and V ∼ L− dr−1. This means that the usual Planck scale becomes an effective coupling constant, describing gravity on scales much larger than the extra dimensions, and related to the fundamental scale via the volume of the extra dimensions:
M 2∼ M 2+d Ld. (5 ) p 4+d
If the extra-dimensional volume is Planck scale, i.e., L ∼ M −p1, then M4+d ∼ Mp. But if the extra-dimensional volume is significantly above Planck scale, then the true fundamental scale M4+d can be much less than the effective scale M ∼ 1019 GeV p. In this case, we understand the weakness of gravity as due to the fact that it “spreads” into extra dimensions and only a part of it is felt in 4 dimensions.

A lower limit on M4+d is given by null results in table-top experiments to test for deviations from Newton’s law in 4 dimensions, V ∝ r− 1. These experiments currently [212] probe sub-millimetre scales, so that

L ≲ 10−1 mm ∼ (10 −15 TeV )− 1 ⇒ M ≳ 10(32−15d)∕(d+2) TeV. (6 ) 4+d
Stronger bounds for brane-worlds with compact flat extra dimensions can be derived from null results in particle accelerators and in high-energy astrophysics [51Jump To The Next Citation Point, 58Jump To The Next Citation Point, 132Jump To The Next Citation Point, 137Jump To The Next Citation Point].
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