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I noticed that there are a lot of similarities between momentum and energy, and it's almost like momentum is the time-like version and energy is the space-like version of the same thing.

For example, energy and momentum are both the integral of force, just energy is with respect to space and momentum is with respect to time. This also explains why they have the same conservation law, because net force is 0 by Newton's laws so the integral of force is constant.

We can also interpret $e=mc^2$ as a statement of equivalence between momentum and energy. $mc$ is in units of momentum, and the other $c$ can be seen as a conversion factor from something time-like to something space-like.

So my question is are there more similarities? Why can energy be converted to different forms while momentum can't?

BioPhysicist
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A. Kriegman
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3 Answers3

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Noether's theorem states that conservation of momentum follows from homogeneity of space (i.e., the invariance of tha physical laws in respect to translations in space), whereas the conservation of energy follows from the homogeneity of time.

In QM this interpretation becomes obvious, since the momentum and the energy operators are just the generators of translations in time and space: $$\hat{p}_j = -i\hbar\frac{\partial}{\partial x_j}, \hat{\mathcal{E}} = i\hbar\frac{\partial}{\partial t}.$$

In the relativity theorey the relationship between the momentum and the energy is also obvious, since the two join together to form the 4-momentum, just like coordinates and time join to form the position in time-space.

Roger V.
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Actually, $E = mc^2$ describes the relation between energy of an object at rest. $E$ is the rest energy and $m$ is the rest mass. The actual relation between energy and momentum in Special Relativity is $$E^2 = (mc^2)^2 + (\vec{p}c)^2$$ Here the $\vec{p}$ is the momentum, and $E$ is the energy of the object. This is the more universal law and applies to all objects, either in rest or in motion (in the rest case, it just collapses to the usual $E = mc^2$).

So, what does this mean? In SR, you take into consideration space-time. So quantities like momentum (usually described in three dimensions of space) are described in 4 dimensions (3 of space and 1 of time). To do this, you need to describe momentum as a four-vector. In the momentum four vector, $E$, the energy is the time component and the other space components form the $\vec{p}$ of the equation above.

So yes, momentum and energy are connected in this way, as energy is the time component of the momentum in $4D$ spacetime.

PNS
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The four momentum:

$$ p^{\mu} = (E/c, \vec p) $$

transforms between frames as a Lorentz 4-vector, so energy (scaled by $1/c$) is the time component and 3-momentum is the space component.

What that means kinematically can be understood by looking at the relationship with 4-velocity

$$ u^{\mu} \equiv (\gamma c, \gamma \vec v) $$

which is:

$$ p^{\mu} = mu^{\mu} =(\gamma mc, \gamma m\vec v)$$

Thus, as an object moves through spacetime with four velocity $u^{\mu}$, the energy is:

$$ E = mu^0c$$

and the 3-momentum is:

$$ p^i= mu^i$$

JEB
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