Surface volume of 3-sphere with radius of 2 Planck length?

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SUMMARY

The surface volume of a 3-sphere with a radius of 2 Planck lengths is calculated using the formula for the hyperarea, which is 2π²r³. This results in a surface volume of approximately 16π² Planck length cubed. The discussion also explores the relationship between this volume and the product of Planck's constant, Einstein's proportionality constant, and Planck time, questioning whether they are equivalent. However, clarity is needed regarding the definitions of area and volume in this context.

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  • Knowledge of mathematical constants such as π
  • Basic principles of dimensional analysis
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mitrasoumya
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What is the surface area ("surface volume") of a 3-sphere having a radius of 2 Planck lengths?

Is the product of the Planck's constant, Einstein's proportionality constant and Planck time also equal to this volume?

Does this equivalence signify anything? What does it signify?
 
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According to wikipedia, the 3-dimensional cubic hyperarea of a 3-sphere of radius r is 2 pi^2 r^3, which should answer your math questions.
 
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Gigaz said:
According to wikipedia, the 3-dimensional cubic hyperarea of a 3-sphere of radius r is 2 pi^2 r^3, which should answer your math questions.

Is the product of the Planck's constant, Einstein's proportionality constant and Planck time also equal to this volume (i.e. where r=2 Planck lengths)?

Does this equivalence signify anything?

What does it signify?
 
I see you are still ignoring what people are telling you about the Planck length. There's no magic to it.

mitrasoumya said:
Is the product of the Planck's constant, Einstein's proportionality constant and Planck time also equal to this volume (i.e. where r=2 Planck lengths)?

Well, you need to make up your mind what you are trying to say. Area? Volume? r? 2r?

mitrasoumya said:
Does this equivalence signify anything?

If you're actually able to get things so the things you are purporting to be equal to actually be equal - something you haven't yet done - you will have proven π = π.
 
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Vanadium 50 said:
Well, you need to make up your mind what you are trying to say. Area? Volume? r? 2r? .
I'll try to reword that. What I am saying is - the product of Planck's constant, Einstein's proportionality constant and Planck time is equal to the "surface" volume of a 3-sphere having the radius of 2 Planck lengths.

Vanadium 50 said:
If you're actually able to get things so the things you are purporting to be equal to actually be equal - something you haven't yet done - you will have proven π = π.
I am sorry I could not understand this part.
 

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