Ink Monitor said:
Just started reading A Brief History of Time, and it has reminded me of a query I have had for years:
E=mc^2
Question: Why does the speed of light have to be squared?
Ultimately it's related to the way that space and time are interrelated.
First some properties of space:
We have pythagoras' theorem.
A two-dimensional space can always be mapped with a grid of perpendicular lines. The grid with its x-axis and y-axis can be oriented in any direction. Take two points in that space, A and B. The grid is in some orientation and we can decompose the distance betwee A and B in component parallel to the x-axis and parallel to the y-axis respectively. Pythagoras' theorem expresses that no matter the orientation of the grid, the distance 'r' between A and P satisfies the rule:
r^2 = x^2 + y^2
This is a principle of
invariance: you can orient the grid (that you are using to map space) in all directions, but the distance between the point is an invariant of that directional freedom.
Pythagoras theorem extends to three spatial dimensions (it extends to any dimensional number)
r^2 = x^2 + y^2 + z^2
Now to the way that
space and time are related.
Special relativity describes that if you measure distance in kilometers and time in seconds then there is an
invariant quantity that is expressed as follows:
\tau^2 = c^2t^2 - x^2 - y^2 - z^2
Here the greek letter 'tau' (\tau) is used for the invariant quantity.
The similarity with Pythagoras' theorem is striking. At the same time, because the spatial dimension has a
minus sign it's completely different from Pythagoras's theorem.
We don't know why that expression \tau^2 = c^2t^2 - x^2 - y^2 - z^2 holds good, but we do know it's profound.
Now, here the factor c
2 is present because of the way the other dimensions are expressed: spatial distance in
kilometers, and time in
seconds. But we are free to express spatial distance otherwise. We can express spatial distance in lightseconds: one lightsecond is the distance that light travels in one second. That way the speed of light is absorbed in the measure of spatial distance. If spatial distance is expressed in lightseconds then the expression for the invariant quantity tau still says the same thing, but without the factor c
2.
In the expression E=mc^2 there is a factor c
2, but that is not a crucial element. The essential thing is that there is a relation of
proportionality between matter and energy. You can opt to express energy otherwise (absorbing the factor c
2 into the energy term), and then the expression would still assert the
proportionality: E=m, which is what that formula all about.
Addendum:
The original question was rather: why a factor c
2, rather than say, c or c
3?
I believe the c
2 factor must arise from the same underlying physics as that phythagoras-like relation between space and time. We don't know
why space and time are related that way. All we know is: our theories are built upon it, and they are very powerful theories indeed, so they must be doing something right.
- I've printed your posts off (my eyes are not too good for staring at the monitor hours on end!) and will read them until it sinks in!
