Dimensions of five-dimensional Newton's constant

In summary, the conversation discusses the dimensions of the five-dimensional Newton's constant in a space with an extra spatial dimension and a compact extra coordinate. The metric and action of the space are also mentioned. The participants compare different definitions and discuss the mass dimension of the Newton's constant in different dimensions of spacetime. The final conclusion is that the Newton's constant has a mass dimension of 2-n in n dimensions and is related to the n-dimensional Planck mass.
  • #1
papades
1
0
Hello,
I have a question, please let me have your answer, if possible:
In a space with an extra spatial dimension , where the extra coordinate is compact:
[itex]0 \le y \le 2\pi \alpha [/itex]
And the metric of the space is:
[tex]{G_M}_N = \left( {\begin{array}{*{20}{c}}
{{g_\mu }_\nu (x)} & {{A_\mu }(x)} \\
{{A_\mu }(x)} & {\varphi (x)} \\
\end{array}} \right)[/tex]

In the action:[tex]S = \frac{1}{{8\pi G_N^{(5)}}}\int {{d^5}x\;\sqrt { - G} \;R} [/tex]

Which are the dimensions of
[itex]G_N^{(5)}[/itex]
, that is five-dimensional Newton’s constant?
Many thanks in advance.
 
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  • #2
Compare your definition to eq. 2.132 here:

http://folk.uio.no/olavau/thesis.pdf

which gives the Einstein-Hilbert action for 4+n dimensional gravity. The prefactor is the 4+n dimensional Planck mass. You'll be able to find the dimension of your 5-dimensional gravitational constant given in units of mass, where c=hbar=1.
 
  • #3
Hmm, the Riemann tensor is esentially a bidimensional object, a set of surface intersections so to say, independently of the number of dimensions of the space time. So in principle it does not sound as a good idea to think of Newton's constant depending of the number of dimensions, it could have more sense to put powers of this constant (and c, when needed) to get the exact dimensionality in each case. But perhaps I have done the same mistake in some of my letters, so I am not to launch the first stone.
 
  • #4
The Ricci scalar has mass dimension 2 for any dimension of spacetime. The Newton constant is defined by the power law at large distances, from which its mass dimension can be determined, or as well as by demanding that the action has mass dimension zero. Therefore, in [itex]n[/itex] dimensions, the Newton constant has mass dimension [itex]2-n[/itex] and is related to the [itex]n[/itex]-dimensional Planck mass by

[itex]G_N^{(n)} \sim \left( M_P^{(n)} \right)^{2-n}.[/itex]
 
  • #5
-3 :smile:
 

1. What is the five-dimensional Newton's constant?

The five-dimensional Newton's constant, also known as G5, is a mathematical constant used in five-dimensional spacetime theories to describe the strength of the gravitational force between two objects.

2. How is the five-dimensional Newton's constant different from the traditional Newton's constant?

The five-dimensional Newton's constant is different from the traditional Newton's constant in that it is used in theories that consider the existence of an additional dimension, while the traditional constant is used in standard three-dimensional theories of gravity.

3. What is the significance of the five-dimensional Newton's constant in physics?

The five-dimensional Newton's constant is significant in theories that propose the existence of more than three dimensions, such as string theory and Kaluza-Klein theory. It helps to explain the behavior of gravity in higher dimensions and provides a more complete understanding of the universe.

4. How is the five-dimensional Newton's constant calculated?

The five-dimensional Newton's constant is typically derived from the four-dimensional Newton's constant, G, using a mathematical equation that takes into account the extra dimension. It is also related to the five-dimensional Planck length and five-dimensional Planck mass.

5. Are there any experiments or observations that support the use of the five-dimensional Newton's constant?

While there is currently no direct experimental evidence for the existence of more than three dimensions, the five-dimensional Newton's constant is used in various theoretical models that have been supported by indirect observations, such as the existence of dark matter and dark energy.

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