mass energy equivilence

JoeB-

Hi

I was just wondering about this..

The formula e = mc2 shows that mass and energy are equivilent... but c2 isn't a 'dimensionless' number.. I know that people say that it is a constant.. but it has a unit, m/s... (or m2/s2?)

So the formula could never be written as e = m even if we redefined the units of mass or energy.. so why is it that they are equivilent?

Or could it be that space and time are themselves equivilent in some respects, and so a unit of 'space over time, (m/s)' cancels itself out somehow to leave a pure dimensionless constant?

Morbert

http://en.wikipedia.org/wiki/Nondimensionalization

We get away with it because we're allowed to ignore dimensions in many cases. So we can often normalise constants like c without feeling guilty.

Delphi91

JoeBallantine wrote: »

Hi

I was just wondering about this..

The formula e = mc2 shows that mass and energy are equivilent... but c2 isn't a 'dimensionless' number.. I know that people say that it is a constant.. but it has a unit, m/s... (or m2/s2?)

So the formula could never be written as e = m even if we redefined the units of mass or energy.. so why is it that they are equivilent?

Or could it be that space and time are themselves equivilent in some respects, and so a unit of 'space over time, (m/s)' cancels itself out somehow to leave a pure dimensionless constant?

If you work out the SI units of each side of the equation, you will find that they are the same.

On the left, Energy is measured in Joules (J). 1 Joule is the work done when a force of 1 newton moves an object a distance of one metre i.e. 1J = 1 Nm. But a Newton is the force required to give a mass of 1kg an acceleration of 1 m/s^2. Putting this all together, we get:

1J = 1kg*(m/s^2)*m
= 1kg (m^2/s^2)

Now look at the right hand side:

m = kg
c = m/s therefore c^2 = (m/s)^2 = m^2/s^2

so mc^2 = kg*(m^2/s^2)

I have always taken the phrase "equivalence" to mean that one can be converted into the other.

Podge_irl

Delphi91 wrote: »

I have always taken the phrase "equivalence" to mean that one can be converted into the other.

Equivalence in its strictest sense (at least physically, rather then mathematically) is a stronger condition than that. "Mass" is quite readily measured in units of energy in particle physics.

Morbert

Delphi91 wrote: »

If you work out the SI units of each side of the equation, you will find that they are the same.

On the left, Energy is measured in Joules (J). 1 Joule is the work done when a force of 1 newton moves an object a distance of one metre i.e. 1J = 1 Nm. But a Newton is the force required to give a mass of 1kg an acceleration of 1 m/s^2. Putting this all together, we get:

1J = 1kg*(m/s^2)*m
= 1kg (m^2/s^2)

Now look at the right hand side:

m = kg
c = m/s therefore c^2 = (m/s)^2 = m^2/s^2

so mc^2 = kg*(m^2/s^2)

I have always taken the phrase "equivalence" to mean that one can be converted into the other.

Yes but when you try to apply the same line of thought to the equation

E = m

you run into trouble. The units no longer match up. But most of the important aspects of physics are expressed as variables and constants. If your unit system is consistent then you don't really need to worry too much about delineations.

Delphi91

Morbert wrote: »

Yes but when you try to apply the same line of thought to the equation

E = m

you run into trouble. The units no longer match up. But most of the important aspects of physics are expressed as variables and constants. If your unit system is consistent then you don't really need to worry too much about delineations.

Could one argue that it's similar to Newton's second law where F is proprtional a and to make them equal, we put in a constant of proportionality, m. Removal of the constant does not imply that F=a.

In Einstein's equation, isn't E proportional to m i.e. the more m you destroy, the more E you create and vice-versa? To "equate" them, the constant of proportionality is c^2. Using dimensional analysis as in my previous post, it can be shown that the units on both sides match up. But that doesn't imply that E=m which is a slightly different proposition (methinks). After all, if E=m then why does E=mc^2??

Thomas_S_Hunterson

Delphi91 wrote: »

After all, if E=m then why does E=mc^2??

When you're not thinking in terms of the 'right' units.

Professor_Fink

Ok, people seem to be losing the run of them selves a little (and not reading the link in Morbert's original reply). Physicists often work in special units such that c=1 and hbar = 1. These simplify a lot of equations, and so it's a handy choice of units for most applications. hbar and c are independant, so we can specify there value separately. In these units, you do get E = m and so units of mass and energy are interchangable, allowing you to for example measure mass in terms of electron volts.

Morbert

Delphi91 wrote: »

Could one argue that it's similar to Newton's second law where F is proprtional a and to make them equal, we put in a constant of proportionality, m. Removal of the constant does not imply that F=a.

In Einstein's equation, isn't E proportional to m i.e. the more m you destroy, the more E you create and vice-versa? To "equate" them, the constant of proportionality is c^2. Using dimensional analysis as in my previous post, it can be shown that the units on both sides match up. But that doesn't imply that E=m which is a slightly different proposition (methinks). After all, if E=m then why does E=mc^2??

I think the confusion lies with the unit system. The base units in the SI system are the metre, kilogram, second etc. From the perspective of relativity, our unit of time (the second) is far too large compared to our unit of distance (the metre) so we get a massive number (c^2) in the energy/mass relation

E = mc^2

But if, instead of using SI units, we used natural units such as the planck length and plack time as base units, we could set the numerical value of c to be 1. We would then be left with the equation

E = m

Now, E and m have different dimensions, but E = m is a dimensionless equation, so we can happily interchange units of mass and energy.