Gravitational Force acting on a massless body

[Lunct]

It's a well known fact that acceleration due to gravity is independent of the mass of the accelerating body, and only depends on the mass of the body it is accelerating towards and the distance from it.
One can prove this mathematically very easily.

F=GMm/r^2 (equation 1)
but also F=ma (equation 2)
so ma=GMm/r^2 so m cancels out and a=GM/r^2 (equation 3)

But what if we are to consider the acceleration acting on a massless object (like a photon) ? Following equation 3, there would still be an acceleration due to gravity, but from equation 1, the product of the masses is zero, and therefore the force would be zero.

So the massless particle would accelerate with zero net force.

Is the problem that by cancelling out the m when it's equal to zero we are effectively dividing by zero?
Or is it that the whole situation is impossible because photons do have a mass from E=mc^2?

And separate follow up question:
I know from briefly glancing Eddington's paper that proved GR in 1919 that there was value for the angle by which light bends by the sun due to Newtonian Gravity. How did they calculate this value?

 

[Dale]

So the massless particle would accelerate with zero net force.

Of course. Massless particles have 0 net force regardless of their (finite) acceleration. $F= ma$

 

[Ibix]

Sean Carroll describes the status of massless particles in non-relativistic physics as "somewhat problematic". Whether light feels a Newtonian gravitational field or not kind of depends on how you hammer the two incompatible theories together. Really you need GR to describe light in a gravitational field - then gravity isn't a force, just spacetime curvature.

 

[anorlunda]

It sounds like this must be what the OP is asking about. Reference to Newtonian gravity is bolded.

https://en.wikipedia.org/wiki/Eddington_experiment
The theory behind the experiment concerns the predicted deflection of light by the Sun. The first observation of light deflection was performed by noting the change in position of stars as they passed near the Sun on the celestial sphere. The approximate angular deflection δφ for a massless particle coming in from infinity and going back out to infinity is given by the following formula:[17]

Here, b can be interpreted as the distance of closest approach. Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio rs/b. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds. This is twice the value predicted by calculations using the Newtonian theory of gravity. It was this difference in the deflection between the two theories that Eddington's expedition and other later eclipse observers would attempt to observe.

 

[Ibix]

As far as I recall you get Eddington's Newtonian deflection figure by plugging an object with initial velocity $c$ into the standard orbital equations and whistling real loud if anyone asks if that's really legitimate. Notably, the object would increase speed on such a trajectory...

 

[vanhees71]

Sean Carroll describes the status of massless particles in non-relativistic physics as "somewhat problematic". Whether light feels a Newtonian gravitational field or not kind of depends on how you hammer the two incompatible theories together. Really you need GR to describe light in a gravitational field - then gravity isn't a force, just spacetime curvature.

"Somewhat problematic."? I'd say it much stronger: In non-relativistic physics there are no massless particles to begin with. The unitary representations of the Galilei group, needed to make non-relativsitic quantum theory consistent with the underlying Newtonian spacetime model, with zero mass don't give dynamics which lead to anything observable in Nature.

Light is in any sense an electromagnetic wave. In the usual astronomical applications (the famous deflection of light on the Sun, first measured by Eddington et al during the eclipse in 1919, predicted correctly by Einstein in 1915) you can use the eikonal approximation (aka geometrical optics) and this can be reinterpreted as if the light rays are trajectories of fictitious massless "particles". Let me stress right away that one should not call these "photons", because that's highly misleading, but that's another story.

 

[Ibix]

"Somewhat problematic."?

The comment is in Carroll's GR notes. The context is that he derives an effective potential for orbits for massive and massless particles in Schwarzschild spacetime, and notes that you can get from the massive to massless case by knocking out one term, and that you can get from the relativistic massive case to the Newtonian one by knocking out a different term. Knocking out both gives you something you might try to interpret as the massless case in Newtonian gravity, which is where the comment appears. I don't think he'd disagree with anything you say - it's a deliberate understatement, I think.

Note that this analysis gives you straight line trajectories, in contrast to the curved paths you get by other approaches. As we've both noted, no Newtonian analysis is likely to be accurate, since Newton is a low speed and weak field approximation to relativity and lightspeed is the definitive high speed.

 

[vanhees71]

If you see it from the point of view of relativity, you also can't do a proper "non-relativistic" limit of the motion of a massless particle in a gravitational field from GR.

The problem is that to get from the geodesic equation of motion of a massive particle to the Newtonian equation of motion you have to assume that the particle's velocity is small compared to the speed of light and that it should stay so when it's moving in the gravitational field, which then also have to be weak, so that you can also approximate the gravitational field by its Newtonian expression.

For a massless particle that procedure fails, because the particle always moves at the speed of light. Of course you can still make the 2nd step, i.e., use the weak-field approximation also in this case, but then you also don't get the naive Newtonian equation of motion, because you need to take into account both the $g_{00}$ and the $g_{rr}$ piece, because the latter cannot be neglected as in the massive Newtonian limit $v \ll c$. That's why the deflection angle is wrong by a factor of 2 when using a naive Newtonian limit for light bending on the Sun.

 

[Dale]

"Somewhat problematic."? I'd say it much stronger: In non-relativistic physics there are no massless particles to begin with.

I disagree. You can certainly have massless objects in classical physics. Massless springs and ropes are common, and students should understand how to use them.

The fact that such objects don’t exist in reality is not terribly important because massless objects in Newtonian mechanics are often a good approximation for real objects with a small mass. All Newtonian physics is an approximation to more exact theories anyway, so wherever a massless Newtonian object is a good approximation for a real object, then it has every bit as much validity as the rest of Newtonian mechanics.

The OP is asking a question in the Classical Physics forum about massless objects in classical Newtonian gravity. The OP points to photons as an example of a massless object. It is therefore reasonable to teach the OP that the photon is not well approximated as a massless Newtonian object, but to make the broad assertion that there are no massless particles in Newtonian physics goes too far, IMO. This is, of course my opinion, and you are free to disagree, but your opinion (stated as a fact) is at least somewhat in conflict with every introductory physics textbook I have ever seen.

 

[PeroK]

The question is of historical relevance. When the deflection of light was measured, there were three possibilities:

1) No deflection, in accordance with what might have been the default position for CM: that light, as EM radiation, would be unaffected by the force of gravity.

2) Deflection based on treating light as a particle subject to Newtonian gravitational acceleration. There may have been no theoretical explanation for this, but it must have been a possibility.

3) Deflection (twice that of 2)) in accordance with GR.

It was historically important that 2) and 3) predict different results. It seems pointless to me to try to analyse this theoretically now. But, in 1919 it was be no means so clear cut theoretically.

 

[vanhees71]

I disagree. You can certainly have massless objects in classical physics. Massless springs and ropes are common, and students should understand how to use them.

This is something completely different. Here you neglect the mass of some parts of a mechanical system against masses of other parts. That's an approximation to a description of a system which can be correctly described by non-relativistic classical mechanics.

The answer to the question of the OP unanimously is that massless particles cannot be described in Newtonian physics, neither in classical nor in quantum mechanics.

In classical relativistic mechanics of course massless particles are mathematically consistently describable. Nevertheless in nature there are no massless point particles.

 

[Dale]

This is something completely different. Here you neglect the mass of some parts of a mechanical system against masses of other parts. That's an approximation to a description of a system which can be correctly described by non-relativistic classical mechanics.

The answer to the question of the OP unanimously is that massless particles cannot be described in Newtonian physics, neither in classical nor in quantum mechanics.

These two paragraphs are contradictory. If “massless particles cannot be described in Newtonian physics” then you cannot use them as “an approximation to a description of a system”.

In classical relativistic mechanics of course massless particles are mathematically consistently describable. Nevertheless in nature there are no massless point particles.

Agreed.

 

[vanhees71]

I never use massless particles in Newtonian physics, because they cannot be formulated.

What you seemed to refer to is the approximation to neglect the mass of parts of a mechanical system that can be described. E.g., if you handle the mathematical pendulum you neglect the mass of the thread the "point mass" with $m>0$ is fastened on. Of course in reality also the thread has a non-zero mass, but it can be neglected compared to the mass of the point mass. This has nothing to do with the fact that massless particles cannot be described in Newtonian physics. There is nothing contradictory here.

 

[Dale]

What you seemed to refer to is the approximation to neglect the mass of parts of a mechanical system that can be described.

Exactly. If you can neglect the mass of part of a system then you can describe massless particles.

Remember, the word particle does not refer to quantum mechanics exclusively, any small object is a particle in Newtonian physics. That was the original meaning of the word. So if part of a system can be described as massless then you can call that part of a system a particle and you have a massless particle.

This has nothing to do with the fact that massless particles cannot be described in Newtonian physics. There is nothing contradictory here.

Here is the contradiction. On the one hand you are saying that you can describe objects as massless in Newtonian physics. On the other hand you are saying that you cannot describe massless particles in Newtonian physics. Newtonian physics does not distinguish between particles and objects. A particle is just a small object. So if an object can be massless and a particle is a small object then a particle can be massless in Newtonian physics.

 

[vanhees71]

Exactly. If you can neglect the mass of part of a system then you can describe massless particles.

No. In my example I describe a massive particle fastened to a thread whose mass is much smaller such that I can neglect it in the calculation. That doesn't mean that I can describe a massless thread in Newtonian mechanics. In fact if you want a precise description you have to include the mass of the thread, and you can measure the (small) difference of the pendulum's frequencies when taking the finite mass of the thread into account.

This has nothing to do with quantum mechanics. I only referred to quantum mechanics to argue, why in Newtonian physics massless particles/objects cannot be described. Forget about quantum mechanics for the rest of the argument.

 

[Dale]

This has nothing to do with quantum mechanics. I only referred to quantum mechanics to argue, why in Newtonian physics massless particles/objects cannot be described. Forget about quantum mechanics for the rest of the argument.

Then your argument is clearly self contradictory.

That doesn't mean that I can describe a massless thread in Newtonian mechanics.

Introductory Newtonian physics textbooks do in fact routinely describe massless threads. So according to the textbooks massless objects can be described with Newtonian physics.

I think the textbooks have it right and you have it wrong here.

 

[PeroK]

Introductory Newtonian physics textbooks do in fact routinely describe massless threads. So according to the textbooks massless objects can be described with Newtonian physics.

It's hard to argue with this!

 

[Lunct]

In classical relativistic mechanics of course massless particles are mathematically consistently describable. Nevertheless in nature there are no massless point particles.

I do not follow. So photons do have mass then?

 

[Ibix]

I do not follow. So photons do have mass then?

No. Photons are massless (as far as we are aware) but aren't point particles. A lot of EM waves have wavelengths measureable in kilometers - there are no point particles in that!

 

[Lunct]

No. Photons are massless (as far as we are aware) but aren't point particles. A lot of EM waves have wavelengths measureable in kilometers - there are no point particles in that!

What is the relation between the wavelength of light and the "size" of a photon? Is it accurate to describe a photon in times of "size"?

 

[vanhees71]

Then your argument is clearly self contradictory.

Introductory Newtonian physics textbooks do in fact routinely describe massless threads. So according to the textbooks massless objects can be described with Newtonian physics.

I think the textbooks have it right and you have it wrong here.

I've never seen described a single massless thread or any other massless body in Newtonian mechanics. "Massless threads" or any other "massless pieces" only occur in the sense of an approximation, where the mass of these pieces can be neglected against other masses, as I wrote above.

I think this is again one of these completely irrelevant discussions about semantics, and we should end it here. It doesn't help to answer the question of the OP.

 

[vanhees71]

I do not follow. So photons do have mass then?

No, photons are massless, but they are relativistic and they cannot be described as particles at all. They are specific states of the quantized electromagnetic field and admit not even the definition of a position observable.

 

[vanhees71]

What is the relation between the wavelength of light and the "size" of a photon? Is it accurate to describe a photon in times of "size"?

No. Photons are not localizable nor have a shape.

 

[valenumr]

No. Photons are massless (as far as we are aware) but aren't point particles. A lot of EM waves have wavelengths measureable in kilometers - there are no point particles in that!

This is probably really wrong as a treatment, but I was thinking about the mass as hf / c². This was just in the context of thinking about different wavelengths of light passing a massive body, and wondering if the huge mass would act like a prism.

 

[ergospherical]

Introductory Newtonian physics textbooks do in fact routinely describe massless threads. So according to the textbooks massless objects can be described with Newtonian physics.

I would say that massless strings/rods/etc. are just a gentler way of introducing a particular constraint imposed on the system, e.g. for a pendulum the constraint that $r \leq L$, or for a rod connecting two particles the constraint that $|\mathbf{r}_1 - \mathbf{r}_2| = L$. It's not actually describing a massless object (which don't exist in the Newtonian theory).

 

[Ibix]

This is probably really wrong as a treatment, but I was thinking about the mass as hf / c².

Energy and mass aren't the same thing. What you are describing here is what is called "relativistic mass", which has been a deprecated concept in professional circles for several decades now. Relativistic mass is the total energy divided by $c^2$ and is a frame dependent quantity (a frame moving with respect to you will see a different $f$ and a different relativistic mass), but mass is frame independent.

 

[valenumr]

Energy and mass aren't the same thing. What you are describing here is what is called "relativistic mass", which has been a deprecated concept in professional circles for several decades now. Relativistic mass is the total energy divided by $c^2$ and is a frame dependent quantity (a frame moving with respect to you will see a different $f$ and a different relativistic mass), but mass is frame independent.

Right, that's why I dismissed the idea. The diffraction would be different for different observes, which doesn't make sense.

 

[jbriggs444]

This is probably really wrong as a treatment, but I was thinking about the mass as hf / c². This was just in the context of thinking about different wavelengths of light passing a massive body, and wondering if the huge mass would act like a prism.

Yes, I believe that this is wrong.

So the notion is that one adopts the [incorrect] model of a photon as a little bullet, takes the [unconventional] notion of mass as relativistic mass, $\frac{E}{c^2}$ and the [well accepted] notion of momentum as $p=\frac{E}{c}$. Then one applies the Newtonian notion of gravitational force and computes the radius of curvature required so that [If I have understood correctly]:$$F = G\frac{m_1 m_2}{r^2} = \frac{dp}{dt} = m_1 \frac {dv}{dt} = m_1 v \frac{d\theta}{dt} = m_1 \frac{v^2}{r} = m_1 \frac{c^2}{r}$$Solving for r:$$r = \frac{Gm_2}{c^2}$$So yes, that leads to a prediction. But not to a prediction that depends on wavelength.

 

[Dale]

It's not actually describing a massless object (which don't exist in the Newtonian theory).

Nonsense. In the many textbooks where massless strings and springs are used they are indeed describing theoretical objects which are theoretically assigned zero mass and then treated according to standard Newtonian theory.

I would agree that massless strings don’t exist in reality, but they are commonplace in Newtonian theory. To state otherwise is to contradict most introductory physics textbooks.

 

[vanhees71]

Ok. Then show an example in a textbook where they write down the equation of motion of a zero-mass object. I don't know a single example. The reason is simply that such a thing cannot even be defined mathematically.

 

[Dale]

Sigh.

https://phys.libretexts.org/Bookshe...Harmonic_Motion/11.02:_Simple_Harmonic_Motion

https://phys.libretexts.org/Bookshe...Waves_in_One_Dimension/12.06:_Advanced_Topics

https://phys.libretexts.org/Bookshe...cs_of_Waves_(Goergi)/05:_Waves/5.04:_New_Page

I know you are well aware of these examples so go ahead with your excuse that, for whatever reason, these examples don’t apply. But these examples all clearly and explicitly involve theoretical massless objects in Newtonian physics.

So your blanket assertion that they don’t exist in Newtonian physics is wrong. You need to make a much more limited claim, something like “massless objects <insert excuse from above> in Newtonian physics”. The broad claim that there are no massless objects in Newtonian physics is refuted.

 

[valenumr]

Ok. Then show an example in a textbook where they write down the equation of motion of a zero-mass object. I don't know a single example. The reason is simply that such a thing cannot even be defined mathematically.

Well, ideal pendelum. I think the point (edit for clarity) [is that] mass is neglected as factor to simplify things, but it's not the same as proper treatment of a massless object.

 

[italicus]

@Dale

I agree with @vanhees71 ;, in Newtonian mechanics, not relativistic, we very often consider some objects as having trascurable mass wrt other parts of the system concerned, and say : let’s put it to zero. For example, dealing with the Atwood machine we in the simplified version trascure the mass of the pulley; but in a more accurate treatment we don’t; this doesn’t mean that the pulley is “massless’ (n the first hypothesis).

 

[vanhees71]

Sigh.

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/11:_Simple_Harmonic_Motion/11.02:_Simple_Harmonic_Motion

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/12:_Waves_in_One_Dimension/12.06:_Advanced_Topics

https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_(Goergi)/05:_Waves/5.04:_New_Page

I know you are well aware of these examples so go ahead with your excuse that, for whatever reason, these examples don’t apply. But these examples all clearly and explicitly involve theoretical massless objects in Newtonian physics.

So your blanket assertion that they don’t exist in Newtonian physics is wrong. You need to make a much more limited claim, something like “massless objects <insert excuse from above> in Newtonian physics”. The broad claim that there are no massless objects in Newtonian physics is refuted.

None of this examples treat the dynamics of a massless system. All are in the category I mentioned from the first posting on this subject on: You neglect the mass of parts of the setup (here the springs) against other masses. Again: There is no physically interpretable dynamics of massless objects in Newtonian physics.

 

[Motore]

The OP already got the answer, but I am posting here another one:
https://physics.stackexchange.com/a/561207

Of course the OP was reminded that the photon cannot be treated accurately using Newtonian dynamics, but a massless object (like a ultra small ball) can be. Here massless means negligible mass and that is what @Dale was referring to all along in my opinion. So a small massless ball does not exist in reality as does not a massless spring, but we can (and we do) use them (as referenced by @Dale in post #31).