## Archive for July 24th, 2022

### It is an entire package: Conservation of energy And conservation of Momentum in same equation. Not just E ≠ mc².

Posted on: July 24, 2022

Yash. July 20, 2022

The Conservation of energy means that no matter what you put in, what reacted, and what came out, the sum of what you began with and the sum of what you ended with would be equal.

But under the laws of special relativity, mass simply couldn’t be the ultimate conserved quantity, since different observers would disagree about what the energy of a system was.

Instead, Einstein was able to derive a law that we still use today, governed by one of the simplest but most powerful equations ever to be written down.

It is, however, odd to note that with this simplicity comes a world of misunderstanding.

E is, in fact, not equal to just mc².

It goes unsaid that Einstein’s energy-mass equivalence is the most famous equation in the world.

Its elegance is unparalleled but you should note that the energy-mass equivalence is not something Einstein went hunting for. It just sort of falls out of the theory of special relativity.

This equation tells you that whenever you’ve got mass, you’ve got energy. But it doesn’t tell you just that. There’s more to it.

What just E = mc² defines for us is that if we’ve got an object of mass, m, then the total energy stored would be equal to the speed of light squared. Whether we can access this energy or not is a question for another day.

Primer, the complete energy-mass equivalence: E is energy, m is mass, c is the speed of light, and p is the momentum.

E is energy, m is mass, c is the speed of light, and p is the momentum.

It’s a little misleading to say that E = mc² is “wrong” and that’s because E is, in fact, equal to mc² when we’re considering the motion of particles to be stationary relative to our frame of reference.

But this is simply the consequence of the p²c² term equaling zero when an object is not moving.

I think it’s quickly apparent why E = mc² is more famous than the longer version of it

It’s correct in most cases we study and it’s a lot more simpler but it doesn’t explain everything.

High school physics introduces the idea that moving bodies have momentum. This plays quite nicely with our equation.

If a body is moving then it must have the energy associated with its mass and the energy associated with its motion. The p²c² term accounts for exactly that.

Here’s the rather interesting part: if we take this equation and consider it for objects that are moving slowly, at “non-relativistic speeds”, we find that we’re met with a lot of complicated terms.

Invariant mass, the first equation. And the binomial expansion of the Lorentz factor: the second equation.

We find that if v is a lot smaller than c, then the fraction v/c is certainly a lot smaller than one. Therefore, this fraction squared is even smaller. To the fourth power and it only gets smaller. Consider the sixth, eighth, and so on.

Point is, the terms are near negligible for non-relativistic speeds.

Neglecting the higher order terms, we can we’re left with just this:

Rather familiarly, this just looks like the kinetic energy term. Well, according to relativity, a moving body also gains these smaller terms of energy but we won’t necessarily talk about that here.

So far, we’ve only considered objects that have mass, are moving, or are stationary.

What happens when we’re dealing with something which does not have a mass? Like photons?

Well, for massless particles, we get something like this: Since the m in the mc² term is equal to 0, we’re met with E² = p²c². Then, just trivially taking the roots on both sides, we get E = pc.

This is true for photons since they transfer energy from one point to another. But of course, with the classical notion of física, we naturally question this too.

Isn’t momentum just the product of mass and velocity and shouldn’t it equal zero for massless particles?

It turns out that momentum is only defined as mv for objects that have mass.

Since its conservation stems from Noether theorem, the maths tells us that the momentum of any closed system is always constant.

The other side of momentum is this: For electromagnetic waves, the momentum is equal to the Planck’s constant h, times the frequency over the speed of light.

Now you could consider the entire Universe as your closed system if you wanted to and you’d find that momentum is conserved.

We can quite accurately measure the change in momentum of bodies as radiation falls on it. This is related closely to the idea of radiation pressure — where light can exert pressure on objects due to the need for momentum to be conserved.

Anyway, we see that E² = (mc²)² + (pc)² successfully describes stationary objects with mass, moving objects with mass, and massless objects like photons. The simplified version E = mc² only really applies to one of these scenarios.

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