# I Modern definition of mass

#### msumm21

If a photon does the 4 things listed below, why do we say it doesn’t have mass? I guess I’m asking if there’s a clear definition of mass.
1. follows geodesics in spacetime (classically: accelerated in a grav field),
2. curves spacetime (classically: creates a gravitational field),
3. has momentum / exerts pressure on things it impacts, and
4. has inertial mass (put it in a mirrored box, now the box’s acceleration decreases per unit applied force).

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#### Ibix

If a photon does the 4 things listed below, why do we say it doesn’t have mass? I guess I’m asking if there’s a clear definition of mass.
Mass in relativity is the modulus of the energy-momentum four vector, which is zero for light (and, indeed, anything else moving on null paths).
follows geodesics in spacetime (classically: accelerated in a grav field),
The status of light in Newtonian physics is rather problematic. However, note that the acceleration due to Newtonian gravity is independent of the mass of the accelerated body. So, even in Newtonian physics, one should not take for granted that massless objects are not affected by gravity.
curves spacetime (classically: creates a gravitational field),
The source of gravitational fields is the stress-energy tensor, which includes mass, but also other things. The gravitational field of light comes from its energy and momentum; it does not need mass.
has momentum / exerts pressure on things it impacts, and
Momentum is not solely a property of things that have mass. Formulae for momentum that depend on mass are only valid for things that have mass.
has inertial mass (put it in a mirrored box, now the box’s acceleration decreases per unit applied force)
Mass is not an additive property (the energy-momentum four vectors add but, unless they are parallel, the modulus of the resultant is not the sum of the moduli of the components), so the fact that a system including light does not have the same mass as the same system with the light removed does not mean that light has mass.

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#### msumm21

Mass in relativity is the modulus of the energy-momentum four vector, which is zero for light (and, indeed, anything else moving on null paths).
OK thanks. I don't know GR yet to understand this, but planning to learn and glad there is an actual definition of this. Probably a reason for it, but it seems strange at first that it's so "inconsistent" with Newtonian physics, in the sense that photons with no mass have most gravity & inertia properties exactly associated with mass before relativity.

#### Ibix

OK thanks. I don't know GR yet to understand this, but planning to learn and glad there is an actual definition of this.
The energy-momentum four vector appears in SR - sometimes called the four-momentum (and Taylor and Wheeler call it the momenergy vector, but I don't think the term caught on widely). It's the source of the famous $E=mc^2$ equation.
it seems strange at first that it's so "inconsistent" with Newtonian physics, in the sense that photons with no mass have most gravity & inertia properties exactly associated with mass before relativity.
The problem you have is that you are trying to carry Newtonian notions "up" into relativity. It's perfectly understandable and we probably all did it at some point, but it'll cause you no end of trouble. The point is that relativity is the more broadly applicable theory - so you need to think about how the relativistic notions simplify into the Newtonian approximations, not wonder why relativity is more complicated than Newton.

#### atyy

If a photon does the 4 things listed below, why do we say it doesn’t have mass? I guess I’m asking if there’s a clear definition of mass.
1. follows geodesics in spacetime (classically: accelerated in a grav field),
2. curves spacetime (classically: creates a gravitational field),
3. has momentum / exerts pressure on things it impacts, and
4. has inertial mass (put it in a mirrored box, now the box’s acceleration decreases per unit applied force).
There is no useful concept of massless particles in Newtonian gravity, and classically light is not treated like a massive particle, but using either ray optics or Maxwell's equations (which don't fit into Newtonian physics with Galilean relativity). In general relativity, the Newtonian ideas of mass generalize into different ideas. A photon has:
1. non-zero gravitational mass (represented by the stress-energy tensor which causes spacetime curvature)
2. zero invariant mass (its dispersion relation is linear, whereas free massive particles have quadratic dispersion relations; massless particles approximately follow null geodesics, whereas free massive particles approximately follow timelike geodesics)
3. indefinite or undefined inertial mass (using the formula valid for massive particles m = mo/√(1-v2/c2) ).

You can also read Stevel Carlip's article "Kinetic Energy and the Equivalence Principle" https://arxiv.org/abs/gr-qc/9909014.

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#### Orodruin

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3. indefinite or undefined inertial mass (using the formula valid for massive particles m = mo/√(1-v2/c2) ).
I would not call this inertia. Inertia in relativity is direction dependent.

#### DrStupid

3. indefinite or undefined inertial mass (using the formula valid for massive particles m = mo/√(1-v2/c2) ).
For v=c the formula is not defined mathematically. But that does't mean that the corresponding physical property is not defined or even infinite. It is clearly given by E/c² - even for massless objects. And I would also not recommend to call it "inertial mass".

#### pervect

Staff Emeritus
Note that the quantity $E/c^2$ depends on the frame of reference, because E, which stands for energy, depends on the frame of reference. The invariant definition of mass does not depend on the frame of reference, but the result is always zero.

By virtue of being independent of the frame of reference, the modern definition of mass, which results in zero for the mass of a photon, is an intrinsic property of a photon that's independent of the choice of reference frame. This also ties in well with Newton's original idea of mass as a "quantity of material", which has the property as being a property of the material and not a property of the combination of the material and the frame of reference.

Popularization of the idea that "energy is mass", related by the famous formula $E=mc^2$, is not compatible with the idea that mass is a "property of matter", because the energy depends on the state of motion - that's rather the whole point!

Let me belabor this point a bit, because I think some readers may be confused about it. (I'm not singling out any specific reader or respondent here, my intent is simply to make some comments based on the numerous times I've seen similar threads in the past).

If a moving baseball has more energy than a stationary baseball (which is true), then energy is not a property of the baseball itself, because one needs to know about both the baseball and what frame of reference it is in to know the energy of the baseball. Put this way, I hope this is obvious - but experience tells me nothing is so obvious it can't be argued about. I probably won't feel terribly motivated to argue much more about this if it doesn't seem obvious to the reader, though.

Photons, unlike baseballs, are never at rest. But the energy of a photon shares the property with the energy of a baseball in that it depends on the frame of reference. In the case of a photon, doppler shift chaanges the frequency of the photon when one considers an identical photon from different frames of reference, and by the relation $E=h \nu$, it also changes the energy of the photon.

Modern professional practice is to talk about the energy and momentum of the photon as frame dependent quantities, often combined into something that's called "the energy-momentum 4-vector", and leaving "mass" as a frame independent quantity that does not depend on the frame of reference. So when one talks about frame-dependent quantites, one talks about energy, and when one talks about frame independent quantities, one talks about mass.

Modern pop-science practice is to talk about energy as if it were interchangable with mass, often resulting in confusion when people do not realize that to know the energy of an object, one needs to know it's frame of reference. And there's no real need to introduce a synonym for energy, mass, when one really means energy. It's clearer just to call energy energy, leaving the term "mass" free to mean something else.

In the full context of general relativity, weighing a photon can give answers that are up to a factor of 2 different from $E/c^2$. But it gets technical, and even a lay discussion involves some long background setting to define exactly what is being measured and how, down to the nitty-gritty of the experimental details. The technically correct answer for the source of gravity in General relativity is that it is not just energy, but it is the stress-energy tensor that causes gravity. To try and makes things as simple and clear as I can while using languge that is not overly technical, I will say that energy, momentum, pressure, and stress can all be sources of gravity in General relativity. I often see resistance on this point, but all I can say is that is the way that General relativity actually works. People are very wedded to the Newtonian idea that it is mass that causes gravity, and nothing else. But General relativity takes a different view - it says that energy, momentum, pressure, and stress all can cause gravity. At some point one has to step beyond Newtonian ideas to understand General relativity.

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