Einstein's curvature vs. Newton's

[Peter Martin]

I've read that observations during the solar eclipse of 1919 showed that Einstein's prediction of light's curvature passing near the Sun was exactly twice that of Newton. I have two questions:

1. Why did Newton assume any curvature? Did he think "light particles" had mass?
2. In the physical world very few things are exactly a given size, mass, speed, etc. Why would Einstein's curvature of light rays be exactly twice that of Newton>

 

[Orodruin]

The gravitational acceleration of a particle is independent of its mass in Newton's theory. It is equal to the gravitational field.

 

[Nugatory]

1. Why did Newton assume any curvature? Did he think "light particles" had mass?

The calculation in question was "If light does have some very small mass, how would it be deflected by a force obeying Newton's $F=Gm_1m_2/r^2$?" Obviously zero mass and hence no deflection was a possible result. The important point is that the prediction of general relativity differs from the Newtonian prediction no matter what assumption we make about the mass of light, so the observation is a test of general relativity either way.

2. In the physical world very few things are exactly a given size, mass, speed, etc. Why would Einstein's curvature of light rays be exactly twice that of Newton>

This exact 2X comes out of the math, which unlike the real world, is exact. We're combining the same numbers (best available values for mass of sun, sun-earth distance, gravitational constant, speed of light, ...) using different formulas, so if those numbers aren't exact we won't get an exact answer for the deflection, but the ratio between the two possible answers will be exactly 2:1. An analogy: I do not know the exact charge of an electron, I cannot measure the distance between two electrons exactly, and I cannot measure the force between two electrons exactly - but I can still use Coulomb's Law ($F=Ce^2/r^2$) to tell me that if I double the distance between two electrons, the predicted force will decrease by a factor of exactly four. If my measurements come out to be 3.999998 or 4.000007 or whatever, that's still consistent with the theory.

 

[Peter Martin]

Why do you say, "in Newton's theory"? Doesn't the feather-coin experiment show that it's true, period?

 

[Peter Martin]

The calculation in question was "If light does have some very small mass, how would it be deflected by a force obeying Newton's $F=Gm_1m_2/r^2$?" Obviously zero mass and hence no deflection was a possible result. The important point is that the prediction of general relativity differs from the Newtonian prediction no matter what assumption we make about the mass of light, so we the observation is a test of general relativity either way.
This exact 2X comes out of the math, which unlike the real world, is exact. We're combining the same numbers (best available values for mass of sun, sun-earth distance, gravitational constant, speed of light, ...) using different formulas, so if those numbers aren't exact we won't get an exact answer for the deflection, but the ratio between the two possible answers will be exactly 2:1. An analogy: I do not know the exact charge of an electron, I can not measure the distance between two electrons exactly, and I cannot measure the force between two electrons exactly - but I can still use Coulomb's Law ($F=Ce^2/r^2$) to tell me that if I double the distance between two electrons, the predicted force will decrease by a factor of exactly four. If my measurements come out to be 3.999998 or 4.000007 or whatever, that's still consistent with the theory.

Thanks for the response. But wasn't Einstein's deflection empirically derived from the photographic plates taken during the eclipse?

 

[Herman Trivilino]

Why do you say, "in Newton's theory"? Doesn't the feather-coin experiment show that it's true, period?

Try replacing the word "theory" with the word "explanation".

 

[Herman Trivilino]

1. Why did Newton assume any curvature? Did he think "light particles" had mass?

I don't know that he ever did assume a curvature! Newton's theory of gravity predicts a deflection. If I understand the history correctly, in preparation for the 1913 eclipse measurements (that never took place) the scientists involved, including Einstein, thought that Newton's theory predicted zero deflection and Einstein's theory predicted 0.875 arc seconds of deflection. By the time of the 1919 eclipse they thought that Newton's theory predicted 0.875 arc seconds of deflection and Einstein's 1.75 arc seconds.

By the way, more modern versions of this experiment use radar ranging, and have confirmed with much greater precision what the results of those eclipse observations confirmed.

2. In the physical world very few things are exactly a given size, mass, speed, etc. Why would Einstein's curvature of light rays be exactly twice that of Newton>

Many things are exact. For example, every electron has exactly the same mass. Every light beam has exactly the same speed. The number of protons in every Helium atom is exactly twice that of the number in every Hydrogen atom.

 

[Nugatory]

Thanks for the response. But wasn't Einstein's deflection empirically derived from the photographic plates taken during the eclipse?

No. It was calculated from Einstein's theory, and then the calculated results were compared with the observed results.

 

[weirdoguy]

Why do you say, "in Newton's theory"? Doesn't the feather-coin experiment show that it's true, period?

In science we call "theory" something that has big empirical support. So he said "in Newton's theory" because the feather-coin experiment show that it's true (whatever "true" means).

 

[jbriggs444]

In science we call "theory" something that has big empirical support. So he said "in Newton's theory" because the feather-coin experiment show that it's true (whatever "true" means).

My take is simpler. Context.

The question was asked: Why does Newton's theory predict a deflection for light? The answer given is that Newton's theory predicts an identical deflection for any massive particle. That Newton's theory is correct for lead balls and feathers is true, but irrelevant. In context, what's relevant is what Newton's theory predicts, not what actually happens.

 

[PeterDonis]

Why did Newton assume any curvature?

He didn't. In fact, Newton himself, AFAIK, never did any calculation like this at all. The "Newtonian" value is a much later calculation, done using Newton's law of gravity and assuming a particle moving at the speed of light just grazing the Sun. Since there is no such thing as a "massless" particle in Newton's theory (more precisely, no distinction between timelike and null worldlines), and no speed of light limit, and since the acceleration due to gravity is independent of the mass of the particle, this calculation can be done and gives a well-defined answer, which turns out to be half the answer given by GR. But the Newtonian calculation does not attribute the effect to "curvature"; it's just the force of gravity in Newton's theory.

Obviously zero mass and hence no deflection was a possible result

The mass of the particle actually drops out of the calculation, and it's not clear to me that "zero mass" makes any physical sense on Newtonian assumptions, since $m = 0$ basically means the thing doesn't exist. But I agree that even if "zero mass" does make sense in this case, it would just predict zero deflection, so either way the result is different from GR.

 

[pervect]

Thanks for the response. But wasn't Einstein's deflection empirically derived from the photographic plates taken during the eclipse?

Derived, no. Detected, yes. Einstein derived his theory before the deflection experiments were done. The deflection experiments were consistent with Einstein's theory, and showed more deflection of light than "Newtonian theory" would predict. The initial tests were not terribly precise, and some argue that they might not, in retrospect, have been definitive. Later much more precise tests have been done, and these later tests are all consistent with EInstein's theory and not consistent with "Newton's theory". I am not aware of any serious claims that the modern results aren't definitive in ruling out Newtonian gravity.

Exactly what is meant by the Newtonian prediction of the deflection of light may be mildly unclear. I'd suggest looking up the <<wiki references>> in the following quote to see what the literature had to say about the Newtonian prediction.

Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object.

For a bit of theoretical background on the 2:1 prediction for light deflection, and on experimental tests of General Relativity, including but not limited to the light deflection tests, I'd suggest reading WIll's "The Confrontation between General Relativity and Experiment", <<link>>

The following excerpt in particular might be helpful in understanding the significance of the equivalence principle, which is also mentioned by the OP to which this post is a reply. The point I'm trying to make is this. The equivalence principle that the OP cites is called the "weak equivalence principle". There is a stronger version, the "Einstein Equivalence principe", discussed by Wills, which can be thought of as the weak equivalence principle, plus local lorentz invaraince, plus local position invaraince, as the article describes.

Here is the full excerpt:

One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called “mass” is proportional to the “weight”, and is known as the weak equivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely falling body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, WEP states that the bodies fall with the same acceleration (this is often termed the Universality of Free Fall, or UFF).

A more powerful and far-reaching equivalence principle is known as the Einstein equivalence principle (EEP). It states that:

1. WEP is valid.
2. The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed.
3. The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed.

The second piece of EEP is called local Lorentz invariance (LLI), and the third piece is called local position invariance (LPI).

I am concerned that an important point of this, the feature of Einstein's equivalence principle called "Local Lorentz invariance", nubmer 2 in the list, may not be clear to the original poster or other non-specialist readers. This feature of General Relativity is an outgrowth of Einstein's previous theory, special relativity. Special relativity has also been well tested, one can find some discussions and references of the tests in the relativity forum's FAQ if one feel the need. I will assume that there is no need and that a fuller discussion of tests of SR would be off-topic. The point I'm trying to make is that Einstein's theoretical framework arose from his desire to make gravity consistent with special relativity, and that the technical language to describe this requirement is "local Lorentz invaraince". Furthermore I want to say that Local Lorentz invariance is a key feature of special relativity. The 2:1 deflection was a mathematical result of Einstein's theory, based on these starting assumptions, and the motivation for this was to make gravity fit into the frameork of special relativity. The prediction of light deflection came after the theory was developed as a way to test the theory. Finally, after Einstein made these predictions, the tests were carried out, and confirmed the theory - or more precisely, the tests were consistent with Einstein's theory, and not consistent with Newton's theory (as described by Soldner).

There are other theories than Einstein's theory, generally more complex ones with additional parameters, that also satisfy Einstein's equivalence principle. Experimental work to test General Relativity is typically done in a framework in which it is assumed that one has a metric theory of gravity, which will satisfy EInstein's equivalence principle. See Will's article for more detail on "metric theories of gravity", and this issue in general.

 

[Orodruin]

But I agree that even if "zero mass" does make sense in this case, it would just predict zero deflection, so either way the result is different from GR.

I am not sure I agree with the zero deflection prediction. You get the zero deflection prediction only if you decrease the gravitational mass while keeping the inertial mass fixed (or going to zero slower). If you want to treat anything like "massless" in Newtonian mechanics you should equate the masses and send them both to zero to get a well defined limit, resulting in the deflection based on the solar gravitational acceleration. As you say, the mass drops out of the acceleration equation, which would mean that any test particle (regardless of mass) starting with the same initial velocity would follow the same trajectory.

 

[Nugatory]

Newton himself did consider the question of gravitational deflection of light - somewhere towards the end of "Opticks" there's a short discussion in which he suggests that light would be deflected by gravity. His thinking, consistent with his corpuscular view of light, seems to have been along the lines suggested by @PeterDonis: zero mass makes no physical sense.

 

[haushofer]

Zero mass in Newtonian dynamics is governed by what we call "Carrollian dynamics". It is kind of an ultrarelativistic Inonu-Wigner contraction on the Poincare algebra, the opposite of the one giving you the Galilei group.

So zero mass gives some sense :P

 

[A.T.]

Zero mass in Newtonian dynamics is governed by what we call "Carrollian dynamics". It is kind of an ultrarelativistic Inonu-Wigner contraction on the Poincare algebra, the opposite of the one giving you the Galilei group.

Mathematically, this specific case of combining Newtonian Graviation with 2nd Law for a test mass of zero is a removable singularity.

 

[Peter Martin]

Thanks to everyone who responded to my query about Einstein's prediction of light's deflection passing near the Sun and Newton's prediction. I'll have to devote some time to understanding the myriad of explanations.

Several responses mentioned Einstein's equivalence principle. I believe that this principle simply says that no experiment can distinguish between the effects of gravity and those of acceleration. How many "equivalence principles" are there?

 

[PeterDonis]

I believe that this principle simply says that no experiment can distinguish between the effects of gravity and those of acceleration.

The more precise version of this is that, within a small local region of spacetime ("small" meaning small enough that tidal gravity is not detectable), no experiment can distinguish between being accelerated in the absence of gravity at some given proper acceleration (such as 1 g), and being at rest in a gravitational field with the same proper acceleration (such as 1 g). This can be generalized to the statement that a small enough region of any spacetime (heuristically, a spacetime containing any kind of "gravitational field") cannot be distinguished by any experiment from a similarly small region of flat spacetime (i.e., a spacetime with no gravity).

How many "equivalence principles" are there?

It depends on what source you read. The Wikipedia article gives a decent summary of the main ones I've seen:

https://en.wikipedia.org/wiki/Equivalence_principle

 

[Riadh]

1-I do not understand why people go on about a photon has zero mass. The mass of the photon can't be measured because of its speed- but it has a mass.. why do we then say that mass and energy are equivalent.. an electromagnetic wave does have momentum, energy and hence a mass- may be a rest mass as well, but we can never tell- as the speed of light is the limit. Standing waves do have a rest mass though.

2-Newton's acceleration is independent of mass and this is pointed out in the very start of this discussion, not to forget the 16th century experiment- Galileo leaning tower of Pisa experiment, showing that bodies fall to ground in the same time regardless of their mass, and as mentioned in most of the answers above. The deflection of light was calculated then using Newton's laws as far back as 1783 and not much later as said, see;

http://adsbit.harvard.edu/cgi-bin/n...75T&defaultprint=YES&page_ind=1&filetype=.pdf

and the final formulae do not have the mass of light in them, only acceleration, plus the mass of the deflecting/gravitating body and the speed of the projectile.

4-Einstein did first calculate the deflection and found it is the same as that of Newton- by the way, he was happy about it, as it confirmed that his theory was correct. Then he revised his calculations after a modification of the theory and found the deflection to be (exactly) twice the value and this 'twice' figure was found to agree with measurement. It was said in the explanation that the first figure comes from 'time' bending and the second comes from 'space bending' which was not considered in the first version of Einstein calculations.

5-I surely do not doubt the sincerity of Einstein results, but finding an exact 'two' factor must make people think why. Surely there must be something that need to be doubled in Newton's results. Since Newton's results do not depend on the 'exact' mass of the photon, and we just showed that a photon have mass, I would conclude that Newton's result need 'somehow' to be multiplied by 2 to agree with Einstein and experiment. I don't see why Einstein and Newton should disagree on this result.. Newton's law is 'empirical', which 'can' be due to 'space time bending' or any other theory, like the 'inverse square', and it does not need the mass of the photon.
6-What is to be explained 'clearly' is the exact doubling of Einstein own results, and how the calculations ended in producing the two equal results. I offered a simple possible explanation which unfortunately got removed by the editors.
See this quote ''the term “rest mass” really only means that the centre of mass of the object is at rest in the frame of the observer'' ;
https://arxiv.org/pdf/1508.06478.pdf

 

[Jonathan Scott]

Provided that the gravitational field isn't extremely strong (as it would be for example near a neutron star) the additional acceleration in General Relativity due to the curvature of space for something moving tangentially relative to a central source works just like motion along a curved line with radius of curvature $r = c^2/g$ where $g$ is the Newtonian gravitational field. As usual, the acceleration moving along such a path is $v^2/r$, so for light it is $c^2/(c^2/g) = g$, so adding on the standard Newtonian acceleration (which can be considered to be due to moving a long a similarly curved line mainly in the time direction) gives a total acceleration of $2g$. For any slower speed, the acceleration due to path curvature in space is $(v^2/c^2) g$ so the total is $(1 + v^2/c^2) g$.

 

[Nugatory]

When we speak of the mass of the photon being zero, we are referring to its rest mass, the value of $m_0$ impied by the relationship between energy and momentum $E^2=(pc)^2+(m_0c^2)^2$. For a photon, we have $E=pc$, so clearly $m_0$ is zero.

The factor of two between the GR and Newtonian predictions is the difference between the predicted trajectory of a particle following a lightlike geodesic in curved spacetime and a particle (regardless of its mass) moving under the influence of an mass-dependent inverse-square force in flat spacetime. Thus, we can and do use observation of the deflection to settle the question of which is a better description of gravity.

 

[Herman Trivilino]

1-I do not understand why people go on about a photon has zero mass.

Because the norm of it's energy-momentum four-vector is zero. In other words, its total energy $E$ is equal to its momentum $pc$. Otherwise it wouldn't travel at the invariant speed $c$.

The mass of the photon can't be measured because of its speed- but it has a mass.

It's a contradiction to claim that a property can't be measured while at the same time claiming to know it exists.

why do we then say that mass and energy are equivalent.

We don't. We say that mass is equivalent to the norm of the energy-momentum four-vector. When the spatial component (momentum) of that vector is zero, then and only then is the mass equivalent to the energy, the so-called rest energy.

an electromagnetic wave does have momentum, energy and hence a mass

The energy equals the momentum, so the difference of their squares equals zero. $mc^2=\sqrt{E^2-(pc)^2}=0$.

Edit: Fixed a missing factor of $c^2$ in the last equation.

 

[Sorcerer]

1-I do not understand why people go on about a photon has zero mass. The mass of the photon can't be measured because of its speed- but it has a mass.. why do we then say that mass and energy are equivalent.. an electromagnetic wave does have momentum, energy and hence a mass- may be a rest mass as well, but we can never tell- as the speed of light is the limit. Standing waves do have a rest mass though.

No, that's only true when speed is zero, as you can see here: E² = (pc)² + (mc²)

Mass equals energy only when p = 0. In the case of light, p = h/λ. When is that going to be zero?