Sorry if this is too basic but we've all got to start somewhere

[Ink Monitor]

Sorry if this is too basic but we've all got to start somewhere...

Just started reading A Brief History of Time, and it has reminded me of a query I have had for years:

E=MCsquared (sorry don't know how to do the little 2 next to the C)

E is Energy, M is Mass and C is the speed of light - Yes? And I can follow that but...

Question: Why does the speed of light have to be squared?

Thanking you
JS

 

[diazona]

A simple answer would be that it makes the units work out. Energy is measured in Joules, and mass is measured in kilograms. One Joule is equal to one kilogram times meter squared per second squared,
$$\mathrm{J} = \frac{\mathrm{kg}\ \mathrm{m}^2}{\mathrm{s}^2}$$
So in order to multiply mass by something to get energy, that something needs to have the units of velocity squared.

That's certainly not an explanation of why $E=mc^2$ is correct, but it should at least help you understand why it's not just, say, $E=mc$.

 

[Cleonis]

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 c² 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 c².

In the expression $$E=mc^2$$ there is a factor c², 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 c² 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 c², rather than say, c or c³?
I believe the c² 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.

 

[Ink Monitor]

Thank you Cleonis and diazona - 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!

 

[userNT369]

I have question about wave frequency of an object.
for example we have 5 meter long wooden stick which is vibrating on frequency of 50mhz. If we change frequency of that wooden stick's some part, let's say 1 meter left part and set its frequency to 100mhz, does this will change whole wooden sticks frequency?
I just want to find out my thoughts if they are true about objects waves. thanks.
I could not find Thread to post there. if there's some general thread questions like this, please refer it :).
I like this forum already )