Question about the Cosmological Constant

[Buzz Bloom]

TL;DR Summary

Reference:
https://arxiv.org/pdf/1203.4513.pdf
Page 4
Equation [2].
PNG image of [2] in main body text.
Using m,s, kg units, all of the terms in the equation, except one, have dimensions m^2/s^2. In order for the term with the Cosmological Constant,Lambda, to also have these units, Lambda must have units 1/m^2. This seems to me to be an oddity.

I am hoping someone can explain to me why the constant Λ has units 1/m².

In the article

https://en.wikipedia.org/wiki/Cosmological_constant​

In the Equations section, the following equation is presented.

Do tensors have dimensions? If so, can someone tell me what the dimensions are for these three tensors: R, g, and T? I know the dimensions for G and c. I am guessing the constant R has the dimension 1/m² since apparently Λ also does.

 

[phyzguy]

Most physical constants have dimensions. G, h, c, ε0, ... all have dimensions. As to the dimensions of R, the metric tensor is dimensionless, as you'll see if you look up its definition. Since the Riemann curvature tensor is derived from second spatial derivatives of the metric tensor, it has units of 1/L^2. Contracting a tensor doesn't change the dimensions, since you are just summing up components. So the Ricci tensor and the curvature scalar also have dimensions of 1/L^2.

 

[Buzz Bloom]

Hi @phyzguy:

Thank you @phyzguy. You have been very helpful.

I have now been able to figure out that the T tensor has dimensions L/M or m/kg. Is this correct?

Regards,
Buzz

 

[phyzguy]

Hi @phyzguy:

Thank you @phyzguy. You have been very helpful.

I have now been able to figure out that the T tensor has dimensions L/M or m/kg. Is this correct?

Regards,
Buzz

No. The stress energy tensor is an energy density, so it has units of Energy/Volume = $\frac{ML^2}{T^2} \frac{1}{L^3} = \frac{M}{L T^2}$. G has units of $\frac{L^3}{M T^2}$, so then GT/c^4 has units of $\frac{L^3}{MT^2} \frac{T^4}{L^4} \frac{M}{L T^2} = \frac{1}{L^2 }$