# Expressing second and its derivatives in terms of seven SI base units

## Main Question or Discussion Point

Time-oriented geometrized units→Second and its derivates in terms of 7 SI base units

I expressed second in terms of meter and kilogram. For doing this I used:

For expressing second in meters:
1s¹ = c = 299792458 m

For expressing second in kilograms:

1s¹ = c³/G = 299792458³/0,0000000000667421 =
= 403703246037718000000000000000000000 kg

How to express ampere, Kelvin, mole and candela in terms of second-derived units, as sketched below?

I know dimension of these seven SI base units in geometrized units as follows:

D ³ →
D ² →
D ¹ → length, mass, time
D º → current, amount of substance, luminous intensity
Dˉ¹ → temperature
Dˉ² →
Dˉ³ →

I got this dimensional analysis by further distilling ST spacetime units from LUFE Matrix from here: http://www.brooksdesign-cg.com/Code/Html/Lm/LMqtySI.htm [Broken] into single geometrized D unit of distance (and its derivatives) along any of timespace dimensions as follows:

|Dˉ³|Dˉ²|Dˉ¹|Dº|D¹|D²|D³| ←D matrix that is simplified version of LUFE matrix

I want to obtain following result:

1sº = ¿ = ? A
1sˉ¹ = ¿ = ? K
1sº = ¿ = ? mol
1sº = ¿ = ? cd

¿ symbol denotes unknown converting formula with constants
? symbol denotes unknown result in target units

Which converting formulas and constants (not rounded, but exact values) use to express these second-derived units in terms of ampere, kelvin, mole and candela?

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pervect
Staff Emeritus
I'm not quite sure what you are doing, but it appears to be roughly similar to geometrized units (see http://en.wikipedia.org/wiki/Geometrized_unit_system) but not exactly the same.

However, geometrized units converts everything to meters (or perhaps cm), so one expresses seconds, kilogramts, etc. in meters. This is done by setting several fundamental constants equal to 1 (c, G and for charge related units, the coulomb force constant).

You might look at how geometrized units handle the cases you mentioned.

Explanation

I'm wanting to do converting every of seven SI base units into seconds (or perhaps daynights) instead of meters, so I want expressing meters, kilograms, amperes, kelvins, moles and candelas in seconds and second-derived units such as seconds to minus one power and seconds to zero power, as I stated above. How exactly to do this? I couldn't find in Internet exactly needed formulas, I found only those for expressing MKS system in terms of seconds. I'm relatively new to physics related things. I didn't made anything physical since 2000 year, and because of losing by me my former physical experience, I need help. My approach differs from Wikipedia geometrized units in choosing second instead of meter, and due to this, Wikipedia formulas doesn't conform to my needs.

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pervect
Staff Emeritus
Here is a very quick answer:

1) converet everything into meters using the wikipedia formulas
2) Multiply the results by 1/c, the conversion factor from meters to seconds. (This has a value of unity since c is assumed to have a value of unity)

i.e.1 meter would be 1 meter x (1 second / 3e8 meters) = 1 meter / c.

Note that c is a constant defined as exactly 299792458.

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Specification

I have problem in transforming from to-meter conversion formulas placed here: http://en.wikipedia.org/wiki/Geometrized_unit_system at Wikipedia into to-second conversion formulas, because of different factors needed, which partially I don't know. For example, to make from Wikipedian kilogram-to-meter formula G/c^2 target kilogram-to-second formula c^3/G, I must multiply it by c^5/G^2. But while transforming second-to-meter c formula back into meter-to-second 1/c formula, I must multiply it by c^-2. As you see, each formula needs different transformation to get from to-meter into to-second conversion. Because to that, I would give from someone final conversion formulas as follows:

1sº = FORMULA = ? A (how many ampers fits in 1sº)
1sˉ¹ = FORMULA = ? K (how many kelvins fits in 1sˉ¹)
1sº = FORMULA = ? mol (how many moles fits in 1sº)
1sº = FORMULA = ? cd (how many candelas fits in 1sº)

I need this in this shape, knowing each exact final conversion formula, because I'm making simpler version of LUFE matrix, and subsequently system of units that is more consistent than SI, that uses only one time dimension instead of two or even seven base units, and I want to recalculate all myself using Windows ME calculator, that supports 32 decimal places, obtaining results in high precision. All needed constants and formulas I got from this PowerPoint presentation: http://web.mit.edu/8.01t/www/materials/Presentations/PPT_W01D3/PPT_W01D3_pc.ppt

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pervect
Staff Emeritus
Google calculator is your friend - or should be. It's quite handy for unit conversions.

G/c^3 is the correct conversion factor from kg into seconds, not c^3/G

This is because G/c^3 has units of

s/kg

Therfore kg * (G/c^3) = kg * (sec / kg) = sec

Examples:

1 kg * (G/c^3) = 2.5e-36 seconds

1 solar mass * (G/c^3) = 5 microseconds

Converting microseconds into meters (by multiplying by the speed of light, c) we get 1500 meters. This is correct, the Schwarzschild radius for 1 solar mass is 1.5 km

c^3/G is the conversion factor to go the other way.

I am assuming you are familiar with the standard pictorial method of unit conversion

N apples * (M oranges / apple) has units of oranges
You chose M so that the factor in the parenthesis is unity.

One must pick certain constants to set equal to unity to perform this conversion, the typical choice is the wikipedia choice, which includes setting the gravitational constant G equal to 1, and the speed of light c equal to 1.

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Gib Z
Homework Helper
Note that c is a constant defined as exactly 299792458.
$$c=299,792,458 m/s^{-1}$$ due to the way meters and seconds have been defined.

$$c=299,792,458 m/s^{-1}$$ due to the way meters and seconds have been defined.
It used to be like that, but now (since 1983) the metre is defined as the distance travelled by light in vacuum during a time interval of 1/299 792 458 of a second[1].

[1] http://en.wikipedia.org/wiki/Meter#Timeline_of_definition

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explanation of c³/G

I used c³/G, instead of G/c³, because I wanted to convert seconds into kilograms, but not kilograms into seconds, which conversion I did from seconds into target units below:

1s¹ = c = 299792458 m
(299792458 meters are contained within 1s¹)
1s¹ = c³/G = 403703246037718000000000000000000000 kg
(403703246037718000000000000000000000 kilograms are contained within 1s¹)

Is this correct?

For these second-derived units listed below I too need four analogous formulas, that converts from second to various powers as below listed, into A, K, mol and cd.

1sº = FORMULA = ? A (how many ampers are contained within 1sº)
1sˉ¹ = FORMULA = ? K (how many kelvins are contained within 1sˉ¹)
1sº = FORMULA = ? mol (how many moles are contained within 1sº)
1sº = FORMULA = ? cd (how many candelas are contained within 1sº)

These units are units derived from second, as I got from simplifying abovementioned LUFE matrix. I want to derive all seven base SI units from second and abovementioned second-derived units, thus I need now only four factors, because first three I already know.

I recently found using Google calculator, that in one second fits 2.62844382 e+75K, because: http://scholar.google.com/scholar?hl=en&lr=&q=(c^5)/(G*k)&btnG=Search. Thus factor from seconds to kelvins is (c^5) / (G * k). How to transform this factor for making converting from reciprocal seconds to kelvins, because of reciprocal second being dimension of temperature?

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Gib Z
Homework Helper
Thats what I meant lol. I was saying c being defined as 299292458ms^-1 is due to the fact the meter is defined and seconds are defined. It is deducible from the definition you posted that c is defined as such.

Thats what I meant lol. I was saying c being defined as 299292458ms^-1 is due to the fact the meter is defined and seconds are defined. It is deducible from the definition you posted that c is defined as such.
Yes and that was my whole point as well.

Help needed

Can anyone, even mentor give me correct conversion factors from second and second-derived units into each of seven SI base units as I described above, and when necessary tell me where I'm wrong in my factors, of course confirming what is good?

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Gib Z
Homework Helper
Yes and that was my whole point as well.
I know and im sorry if you got the impression I thought you didnt know, its just that I run into alot of people who think its amazing that c is exactly 299792458, they wont think its so amazing when they find out the definitions...

Conversion

How to transform (c^5)/(G*k) expressed in K/s to make it expressed in K*s that is equal to K/Hz ? I now need to know how many kelvins fits into one hertz.

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conversion from hertzes into seconds solved

I found that to get how many kelvins fits into one hertz I must use formula h/kB, which tells me that into one hertz fits 0.00000000004799237 K.

Proof here:

Good conversion table suitable for my task is here:
http://ptf.fuw.edu.pl/stale/stale_fiz.pdf [Broken] - please look at nearly lowest section, where is header with J kg mˉ¹ Hz and K eV u Eh symbols. It helped me very much, because of lack of any further help from any forum members.

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rbj
It used to be like that, but now (since 1983) the metre is defined as the distance travelled by light in vacuum during a time interval of 1/299 792 458 of a second[1].

[1] http://en.wikipedia.org/wiki/Meter#Timeline_of_definition
but i think the point is the need of units attached to the 299792458 number for c. in SI, c is not dimensionless, even if it is defined constant.

if it were Natural Units (Planck or something else) then it is meaningful to define the speed of light in vacuo as either 1 (Planck units and most other systems) or sometimes as $\alpha^{-1}$ = 137.03599911 (Atomic units). so it can make sense that c can be either of those two dimensionless numbers, but not the dimensionless number: 299792458.

Fredrik
Staff Emeritus
Gold Member
$$c=299,792,458 m/s^{-1}$$
I'm waiting for a pot of water to boil, so I'll just nitpick some silly mistakes until the water's ready for my spaghetti. I'm sure you didn't really mean meter-seconds, so you should have written $ms^{-1}$ or $m/s$.

but i think the point is the need of units attached to the 299792458 number for c. in SI, c is not dimensionless, even if it is defined constant.

if it were Natural Units (Planck or something else) then it is meaningful to define the speed of light in vacuo as either 1 (Planck units and most other systems) or sometimes as $\alpha^{-1}$ = 137.03599911 (Atomic units). so it can make sense that c can be either of those two dimensionless numbers, but not the dimensionless number: 299792458.
I don't see the disagreement here. I've never claimed c was dimensionless: only in units where displacements in space and displacements in time are measured by a common unit (sometimes they are) is c dimensionless.

I saw that between h/kB [K/Hz] and (c^5)/(G*k) [K/s] exist inconsistency in converting, because if 1Hz=1s, then conversions should be the same, but I give http://www.google.pl/search?hl=en&q=h/k&btnG=Szukaj&lr= and http://www.google.pl/search?hl=en&q=(c^5)/(G*k)&btnG=Search that are not even their reciprocals. Because of this mish-mash in formulas, I need from anew help in converting seconds and second-derived units into meters, kilograms, seconds, amperes, kelvins, moles and candelas as follows:

1s¹ = FORMULA = ? m (how many meters fits in 1s¹)
1s¹ = FORMULA = ? kg (how many kilograms fits in 1s¹)
1s¹ = FORMULA = ? s (how many seconds fits in 1s¹)
1sº = FORMULA = ? A (how many ampers fits in 1sº)
1sˉ¹ = FORMULA = ? K (how many kelvins fits in 1sˉ¹)
1sº = FORMULA = ? mol (how many moles fits in 1sº)
1sº = FORMULA = ? cd (how many candelas fits in 1sº)

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rbj
I don't see the disagreement here. I've never claimed c was dimensionless: only in units where displacements in space and displacements in time are measured by a common unit (sometimes they are) is c dimensionless.
no big deal. it's just that GibZ corrected a possible oversight of MeJennifer (i thought correctly, c is not simply the number 299792458) and i thought you were correcting GibZ. i dunno. i'll just watch.

I made working formula from seconds to kilograms in form:
((1 second)*(c²/h))-¹
Both second and (c²/h) are to -¹ power.

Thus I transformed it to:
(1 second)-¹*(c²/h)-¹

But how to transform it further to make bare (1 second) on left side of multiplication (or division if needed) sign, and appropiate transformation on right side of multiplication (or division if needed) sign, that too converts from seconds to kilograms, and has only one instance of second on left?

[c] = L T-1
[h] = M L2 T-1

[c2/h] = L2 T-2 M-1 L-2 T1 = M-1 T1 = T/M.

Therefore, in SI units:

1 second = c2/h * 7.372 -51 kg
1 kg = h/c2 * 1.356 x 1050 second

EDIT: oops! forgot to put in the conversion factors.

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[c] = L T-1

1 second = c2/h * 7.372 x 10-51 kg
1 kg = h/c2 * 1.356 x 1050 second
I deduced that these factors:

7.372 x 10-51
1.356 x 1050

are derived from:
h/c2
and
c2/h

but unit dimension doesn't match, because of introducing of dimensionless unit made by multiplication by inversion. Thus how finally these two expressions:

1 second = c2/h * 7.372 x 10-51 kg
1 kg = h/c2 * 1.356 x 1050 second

are formulated using combination of constants only?

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You can't do those without constants in the SI system: these factors are not chosen so that they are normalised. Of course, you are free to use any other standard.