# A photon's energy equals its spatial momentum?

## Main Question or Discussion Point

This is taken from B. Schutz's First Course in General Relativity.

Am I alone in considering this an abuse of language? After all, energy and spatial momentum are NOT the same concept, although they may be both expressed in kg in the units used (in which c=1).

I would rather say that a photon's energy numerically equals their spatial momentum in units such that c=1, but not in any other units.

I am puzzled because I consider Schutz's text quite rigorous.

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PAllen
2019 Award
Please give us the full quote in context.

Sure.
"if a photon carries energy E in some frame, then in that frame p0 = E. If it moves in the x direction, then py = pz = 0, and in order for the four-momentum to be parallel to its world line (hence be null) we must have px = E. This ensures that

$\vec{p}\cdot\vec{p}$ = -E2 + E2 = 0​

So we conclude that photons have spatial momentum equal to their energy"

PAllen
2019 Award
I think, in context, it is clear he means "in appropriate units". However, I would not argue that a more universally clear statement would be "proportional', and that the metric in the given convention ensures that norm of the 4-momentum is zero.

Meir Achuz
Homework Helper
Gold Member
Energy and momentum are components of a four-vector. Any system that gives them different units is inconsistent
(even if widely used).

PAllen
2019 Award
Energy and momentum are components of a four-vector. Any system that gives them different units is inconsistent
(even if widely used).
That's not true. A 3-vector's components in spherical coordinates has two dimensionless angles and a distance. The metric 'fixes this up' as it were.

Energy and momentum are components of a four-vector.
Energy/c and momentum are components of a four-vector and have both the same unit not only for c=1.

Energy/c and momentum are components of a four-vector and have both the same unit not only for c=1.
Exactly. The first component of the four momentum is E/c, while the spatial components are spatial momentum, px, py, pz. In SI they are all expressed in Kg.m/s while in geometrized units they're all kg. You can only say that energy equals momentum for photons "numerically" (in some unit system). It's like saying that for a 1m-sided square, the side equals the surface. It has no physical meaning.

I guess what I mean is that geometrized units, for all their convenience, have to be handled with extreme care as to not create conceptual confusion in the minds of newbies like me.

WannabeNewton
Well for starters Schutz' text isn't rigorous by any means (not that that's a bad thing). Secondly, to a relativist and physicists in other areas of physics (e.g. HEP), natural ("God given") units are basically the only units in existence when doing theoretical and conceptual calculations-the usual units are only useful when having to compute numerical values for quantities or when having to make order of magnitude estimates.

If you're just starting out the confusion is definitely understandable but it's best to get used to it now rather than later because interchangeably using energy and mass for time-like particles and momentum and energy for light-like particles will be immeasurably ubiquitous in the literature.

Lastly, saying "it" has no physical meaning is entirely incorrect. It actually has more physical meaning than what you're used to in the god-awful SI units.

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jtbell
Mentor
Note that particle physicists talk about both momentum and mass in terms of energy all the time. For example, an electron with mass 511 keV/c2 and momentum 600 keV/c has energy √(5112 + 6002) = 788 keV.

They commonly even say in conversation among themselves things like "the mass is 511 keV" and "the momentum is 600 keV", with the unit-factors of c being understood as present.

Lastly, saying "it" has no physical meaning is entirely incorrect. It actually has more physical meaning than what you're used to in the god-awful SI units.
That bit got me thinking. When I first encountered geometrized units, I thought they were just convenient units so that c and c2will disappear from the equations. But it seems that accepting geometrized units means accepting that momentum and energy are really the same concept, both measured in kg and therefore directly comparable.
What makes me wonder if the time between two consecutive beats of my heart is bigger or smaller than the distance between New York and Paris (it's bigger, of course) and what that actually means.

That bit got me thinking. When I first encountered geometrized units, I thought they were just convenient units so that c and c2will disappear from the equations. But it seems that accepting geometrized units means accepting that momentum and energy are really the same concept, both measured in kg and therefore directly comparable.
What makes me wonder if the time between two consecutive beats of my heart is bigger or smaller than the distance between New York and Paris (it's bigger, of course) and what that actually means.
What are units anyways?

Meir Achuz
Homework Helper
Gold Member
c is huge in SI, but 1 in natural units. Is it large or small?
e ~10^-19 in SI, but ~1/10 in natural units. Which is the real e?

I am not concerned with the magnitude of the units.

The way I see it E=pc is true in whatever units you choose to measure time, while E=p is only true if time is measured in meters. I always thought that the truth of an equation in physics should not depend on the units chosen to represent the magnitudes.

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Meir Achuz
Homework Helper
Gold Member
E and p can be measured in any consistent unit. Only, if they are measured in inconsistent units is a conversion constant needed. For the distance, NY-London, if you have an appointment in London, airline hours is the most useful distance unit. It would be awkward to use heartbeats for that distance unit, but you could if you wanted to.
The SI distance unit, meters, would be even less useful.

Nugatory
Mentor
The way I see it E=pc is true in whatever units you choose to measure time, while E=p is only true if time is measured in meters. I always thought that the truth of an equation in physics should not depend on the units chosen to represent the magnitudes.
The question comes down to asking whether it's reasonable to consider measuring time in different units than distance. It would be perverse to represent distances along the ground in meters and heights above the ground in yards - so we don't and we're happy to write equations whose correctness depends on choosing the same units for both.

So why aren't we just as happy with $E^2=m^2+p^2$?

One reason is our intuition that using different units for time and distance is not as perverse as using different units for distance and height; deep down in the prehistoric lizard-brain part of our minds we think that they really aren't the same (and the signature of the metric agrees). Another reason is that we've filled the world with measuring devices that use different units for space and time, and a third reason is that although meters are well-suited to human perceptions of distance, they're silly small for human perceptions of time.

I don't see that any of the reasons should apply in a physics textbook, but there's an element of personal taste at work here. I am just as happy (actually, happier) with $E^2=m^2+p^2$ rather than $E^2=m^2c^4+(pc)^2$ but there's no reason why you have to share this preference.

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That probably depends on where you want your physics to evolve. Remember that the principal equation of Wave Mechanics which de Broglie discovered is $v=\frac{\partial E}{\partial p}$

Setting $c=1$ is definitely sweeping certain things under the carpet, such as the fact that in Wave Mechanics
vphase vgroup = c2

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