A Tiny Error in All Objects and Some Questions

 

In The Mass of Everything, I referred to the paper All objects and some questions and most specifically the image below:

That figure had a long text below it which includes the following statement: “The smallest possible object is a Planck-mass black hole indicated by the white dot labeled ‘instanton’ (Ref. 20). Its mass and size are (m,r)=(m_P,l_P).”  Ref 20 is a paper by Carr and Rees called “The anthropic principle and the structure of the physical world” published by Nature in 1979 and not easily accessed.  I do note that Carr wrote a later paper, Does Compton / Schwarzschild duality in higher dimensions exclude TeV quantum gravity?, in 2018 which includes this image:

The paper also includes this statement: “The Compton and Schwarzschild (radius) lines intersect at around the Planck scales, R_P = √ħG/c³ ≈ 10⁻³³cm, M_P = √ħc/G ≈ 10⁻⁵g”.  Note the lack of precision in Carr’s paper.  This is, as it turns out, fully justified.

The Schwarzschild radius is given by r_S=2GM/c², so where the radius is the Planck r_Pl=√ħG/c³ length we get:

r_S=2GM/c²=√(ħG/c³)

So,

M=√(ħG/c³).c²/2G=√(ħc/G)/2=m_Pl/2

Therefore, the instanton must be (m,r)=(m_P/2,l_P) or, possibly, (m,r)=(m_P,2l_P), but it cannot be (m,r)=(m_P,l_P).

Alternatively, Lineweaver and Patel could have written “The smallest possible object is in the range of a Planck-mass black hole indicated by the white dot labeled ‘instanton’ (Ref. 20). Its mass and size are of the order of the Planck mass and length.”  And the chart would have to be updated to put approximations against the m_P and l_P that intersect on the instanton.

Or, and this would be more appropriate, put a “/2” after m_P in both instances and reword to “The smallest possible black hole has a Planck-length radius indicated by the white dot labeled ‘instanton’ (Ref. 20), and mass of one half of a Planck-mass.”