Exploring the equivalences between mass, time and length

We know that in the universe we can establish an equivalence between length and time through a constant (the speed of light): l = ct and talk about length in units of time (light-years).

But, by playing mathematically with the dimensions of M, L and T and the constants G and c we can see with some ease that we also unite some relationships between mass and time and mass and length (or radius) through a constant.

Are the following:

m = (c^3/G) t and m = (c^2/G) l (m=mass, t=time, l=lenght)

(so we could measure mass in units of time or length)

If we provisionally call these constants Y = c^3/G and Z = c^2/G , these constants have specific values:

Y = 4.037 x 10^35 Kg/sg

Z = 1.347 x 10^27 Kg/m

(that is, very large masses correspond to a second and a meter) (if no other numerical constant appears)

Do these constants Y and Z have a name in physics?

These constants could also be expressed in the following units:

Y = 4.037 x 10^35 newton/c

Z = 1.347 x 10^27 newton/c^2

since we could also write these identities as m = (Fp/c) t and m = (Fp/c^2) l (where Fp is the Planck force)

What physical meaning could these equivalences have?

If they had a physical meaning, we should actually add a possible dimensionless numerical constant k (real number) and write:

m = kYt & m=kZl (or m = kπZr if length is a radius)

That is, the question would be whether these equations can have any physical meaning.

Could we say, looking at these equations, that mass (i.e. matter) is “just” a certain manifestation of space-time?

The system in which ##c=G=1## is usually called "geometrical units". They aren't universally used, even in relativity where ##c=1## is absolutely standard. I think it's because ##c=G=1## implies ##\hbar\neq 1##, and you eventually have to decide which one you prefer (or just stick with kilograms and not worry about it).

Unit choice doesn't mean anything at all. Setting ##c=1## makes a lot of sense because the Lorentz transforms imply that space and time can be "rotated" into each other to an extent, so it makes sense to use the same units for both. Mass has no such obvious relationship to space or time, which is probably why geometric units aren't as popular as ##c=1## alone.

Here is the Wikipedia page on geometrized units:

https://en.m.wikipedia.org/wiki/Geometrized_unit_system

As far as physical significance, it is just another unit system. It is convenient for doing general relativity but less convenient for buying bacon. In general relativity there is a close relationship between mass (stress energy) and curvature, so this makes particular sense there. But you can use geometrized units even where it doesn’t make particular sense.

mister i said:

What physical meaning could these equivalences have?

The same physical meaning as there being 12 inches in a foot or 10 dimes in a dollar. None at all. Unit conversions are just convention.

Sorry, but in this post I am not talking about unit conversions, but rather I deduce "virtual" equations solely through constants and dimensions. Specifically, these are the 4 that I put here again:

(1) ## m = k (c^3/G) t ##
(2) ## m = k (c^2/G) l ##
(3) ## m = k (Fp/c) t ##
(4) ## m = k (Fp/c^2) l ##

Where Fp is the Planck force and k a possible real number that can be 1 or not.

Imagine an amateur physicist at the end of the 19th century who reads in a (news)paper that Maxwell's equations imply that the speed of light is constant. Using simple concepts of classical physics, constants and dimensions he could deduce the following:

## E= W =F l ## and ## F = m a ##

so, using dimensions:

## E = (M) (L/T^2) (L) = M (L/T)^2 ## which are the dimensions of ## m c^2 ##

So, if there were physics forums at that time, this person could have asked: Only by using dimensions I have arrived at a curious relationship between energy and mass:

## E = (k c^2) m ##
where k is a possible number that can be 1 or not.

He asks: Could this have any physical meaning? (remember that we are at the end of the 19th century) and a physicist of that time answered: "There is no physical meaning at all. Unit conversions are just convention"

But now, returning to the present, I realize that in the case of equation (2), if ## m = E/c^2 ## then equation (2) becomes:

## m = E/c^2 = k (c^2/G) l##

it is to say

## E = k (c^4/G) l ##

If we give k the value of ## 1/8π## , we obtain:

## E = (1/8π)(c^4/G) l## or, changing the order ## l = (8π G/c^4) E##

which is a pseudo-proto-simplification of the GR (as advanced by Dale who has suggested a possible physical meaning of one of them).

In fact, I am intuitively convinced that any equation where the dimensions of the two sides of the = sign coincide must have some hidden physical meaning.

mister i said:

In fact, I am intuitively convinced that any equation where the dimensions of the two sides of the = sign coincide must have some hidden physical meaning.

That's what it's all about!

You are just doing dimensional analysis. You can find many relations between various quantities this way, and most will be meaningless. You are cherry picking results that are similar to ones with which you are familiar.

mister i said:

Sorry, but in this post I am not talking about unit conversions, rather I deduce "virtual" equations solely through constants and dimensions

Yes, you are. You are talking about the equivalence between mass and length. That is geometrized units. And even your process of deducing the equivalence through constants and dimensions is the same as geometrized units.

It is cool that you figured it out on your own, but what you figured out is in fact already known and it is called geometrized units.

mister i said:

(1) ## m = k (c^3/G) t ##
(2) ## m = k (c^2/G) l ##
(3) ## m = k (Fp/c) t ##
(4) ## m = k (Fp/c^2) l ##

Where Fp is the Planck force and k a possible real number that can be 1 or not.

Did you look at the wikipedia link? Your first two values come right off the table there:

And the second two can be obtained from the first two with the additional conversion factor

mister i said:

some hidden physical meaning

What do you think about this one:

"It is important to realize how profound Planck’s insight was. Nature herself, far transcending any silly English king or some self-important French revolutionary committee, gives us a set of units to measure her by. We have managed to get rid of all manmade units. We needed three fundamental constants, each associated with a fundamental principle, and we have precisely three! This suggests that we have discovered all the fundamental principles that there are. Had we not known about the quantum, then we would have to use one manmade unit to describe the universe, which would be weird. From that fact alone, we would have to go looking for quantum physics."
Zee, A. Einstein Gravity in a Nutshell