Reframing Light Bending Using Density Instead of Radius

[Owe Kristiansen]

TL;DR

Title: Reframing Light Bending Using Density Instead of Radius

I’ve been exploring an alternative way to express gravitational light bending:

θ = (4G / c²) * (4π / 3)^(1/3) * M^(2/3) * ρ^(1/3)

Instead of using radius r, I substituted it using the density ρ of a spherical mass.

Title: Reframing Light Bending Using Density Instead of Radius

I’ve been exploring an alternative way to express gravitational light bending, starting from the classical general relativity approximation:

θ = (4 * G * M) / (r * c²)

Instead of using radius r, I substituted it using the density ρ of a spherical mass. It should be correct at the radius of a sphere with uniform density.

For a sphere:

ρ = M / V = M / ((4/3) * π * r³)

Solving for r gives:

r = (3M / (4πρ))^(1/3)

Substituting this into the original bending formula gives:

θ = (4G / c²) * (4π / 3)^(1/3) * M^(2/3) * ρ^(1/3)

This reframing expresses light bending as a function of mass and density, rather than radius. It seems to offer a more material-based interpretation of curvature — compactness and mass together determine how much light bends.

The beauty of this is that more mass or more density increase the bending.

I’m curious whether this formulation has been explored before, or if it might offer any new insights into gravitational lensing or curvature fields. Would love to hear your thoughts.

 

[Ibix]

Unfortunately you're assuming Euclidean geometry, and the whole basis of general relativity is non-Euclidean geometry. Very few astrophysical objects have uniform density and a great many of them aren't spherical.

Also, the total mass of a compact object (or at least the mass times G) is easy to measure from orbits of nearby objects. The density and surface area may be unknown (and, indeed, the density of a black hole isn't a well defined concept). So I don't see that this is a more convenient form of the equations even in the kind of weak field approximation where it's valid.

 

[Owe Kristiansen]

Unfortunately you're assuming Euclidean geometry, and the whole basis of general relativity is non-Euclidean geometry. Very few astrophysical objects have uniform density and a great many of them aren't spherical.

Also, the total mass of a compact object (or at least the mass times G) is easy to measure from orbits of nearby objects. The density and surface area may be unknown (and, indeed, the density of a black hole isn't a well defined concept). So I don't see that this is a more convenient form of the equations even in the kind of weak field approximation where it's valid.

Thanks for reply, and yes, it may have limited practical usage, but may be useable in corner cases.

The reason I dig into it is I gain understanding by studying ratios.

 

[PeroK]

Thanks for reply, and yes, it may have limited practical usage, but may be useable in corner cases.

The reason I dig into it is I gain understanding by studying ratios.

The deflection angle depends on the impact parameter of the light, not the radius of the gravitating mass.

 

[Owe Kristiansen]

The deflection angle depends on the impact parameter of the light, not the radius of the gravitating mass.

Thanks,

In the specific case I was looking at — light just grazing the Sun — the impact parameter is equal to the Sun’s radius. So using r = R isn’t a mistake, it’s modeling the real-world situation that was actually measured during the 1919 eclipse.

What I found interesting is that the full general relativity result:

phi = 4GM / (rc²)

can be written as a simple energy ratio:

phi = (2 × gravitational potential energy) / (relativistic kinetic energy)
phi = (2GMm / r) / (½ mc²)

This gives the correct 1.75 arcseconds when r is the Sun’s radius.

 

[PeroK]

There was a thread about this a couple of years ago. Someone was trying to justify (intuitively) why the GR approximation gave twice the hypothetical Newtonian calculation. You might be able to find the thread.

It was largely a waste of time, IMO.

 

[PeroK]

Here it is:

Undergrad Thread 'The bending of starlight is twice the Newtonian prediction'

May 31, 2021

When light is viewed from an accelerating frame of reference it appears to bend at a rate of a/c (the acceleration a being at right angles to c). By the Principle of Equivalence the bending of a beam of light in a uniform gravitational field ought to be g/c rad/sec. In principle you could use this expression to calculate the total bending of light as it passed a massive object by fairly straightforward integration. I call this the 'Newtonian' prediction.
I am told that GR predicts the bending of starlight round a massive object to be twice this 'Newtonian' prediction. Does this mean that...

• J O Linton
• Bending Newtonian Prediction
• Replies: 90
• Forum: Special and General Relativity

 

[Owe Kristiansen]

There was a thread about this a couple of years ago. Someone was trying to justify (intuitively) why the GR approximation gave twice the hypothetical Newtonian calculation. You might be able to find the thread.

It was largely a waste of time, IMO.

One try?

When you integrate or derivate formulas factors appear, so my hypotesis is I think the factor 2 is a result of that. 2 / 1/2 = 4, which is the conatant that appears in the standard phi formula, 4GM / rc²

The constant 4 in (4GM / rc²) becomes 2 when you introduce 1/2 mc^2 in the denominator according to math rules.

 

[weirdoguy]

according to math rules.

But we are doing physics, and newtonian physics does not work in this context.

 

[Owe Kristiansen]

But we are doing physics, and newtonian physics does not work in this context.

Ok, I will dig into that.So, what is the formulas replacing newton with something better?

 

[PeroK]

Ok, I will dig into that.So, what is the formulas replacing newton with something better?

General Relativity.

 

[Ibix]

Ok, I will dig into that.So, what is the formulas replacing newton with something better?

As PeroK says, you need to study general relativity. If you look for early tests of GR you'll find Eddington's measurement of the deflection of light grazing the Sun, which tested Einstein's predictions. The derivation of the deflection angle will be there.

Note that potential energy is of very limited use in GR. It can't be defined in anything but the simplest of circumstances.

 

[Owe Kristiansen]

General Relativity.

So, the formula for the bending of light in general relativity is:

phi = 4GM / (rc²)

So that is the one I am using. Execpt some syntactic sugar where other letters are used, meaning the same. r vs b

So, then I can use math to adjust this formula.

phi = (4GM) / (rc²)

multiply by 1 -> (1/2m)/(1/2m):
phi = ((1/2m) 4GM) / ((1/2m)rc^2)

Rearranging and moving r up as denominator:
phi=(((1/2*4GMm/r) ) / (1/2mc^2)

And we have arrived on
phi= 2 * (GMm/r) / (1/2mc^2)

This is the exact same formula as we started on, so its still general relativity.

 

[PeroK]

So, the formula for the bending of light in general relativity is:

phi = 4GM / (rc²)

That's only an approximation (for small $GM/rc^2$). Note that in some cases, Newtonian gravity is a close approximation to GR - orbits of the planets round the Sun, for example. In this case, it isn't even close. You want to make a big deal that the difference is almost exactly a factor of 2. That may just be a numerical coincidence.

What we can say is that Newtonian gravity cannot predict, even approximately, the quantitative bending of light round the Sun. The comparison between GR and Newtonian gravity in this case, therefore, has little or no value.

 

[Owe Kristiansen]

That's only an approximation (for small $GM/rc^2$). Note that in some cases, Newtonian gravity is a close approximation to GR - orbits of the planets round the Sun, for example. In this case, it isn't even close. You want to make a big deal that the difference is almost exactly a factor of 2. That may just be a numerical coincidence.

What we can say is that Newtonian gravity cannot predict, even approximately, the quantitative bending of light round the Sun. The comparison between GR and Newtonian gravity in this case, therefore, has little or no value.

Ok, I am looking at the general and special relativity formulas now.

 

[weirdoguy]

Don't look at the formulas, look at the whole story!

 

[Owe Kristiansen]

The formulas I have used are for weak field approximations. The thread is also tagged with gravity and newton. So, while special and general relativity is another liga, the weak field approximation predicts well enough for the bending of light near our own sun.

 

[Ibix]

So, while special and general relativity is another liga, the weak field approximation predicts well enough for the bending of light near our own sun.

Newtonian gravity and the weak field approximation are different things. Newtonian gravity has the additional constraint that nothing is moving quickly, which rules out dealing with light.

That's why you have to add fiddle factors to your formulas to get the right answer - they aren't valid for the regime you want to study.

 

[Owe Kristiansen]

Newtonian gravity and the weak field approximation are different things. Newtonian gravity has the additional constraint that nothing is moving quickly, which rules out dealing with light.

That's why you have to add fiddle factors to your formulas to get the right answer - they aren't valid for the regime you want to study.

Ok, ok. You’re right that Newtonian gravity and the weak-field approximation in general relativity aren’t the same—and I appreciate the clarification.

I fully acknowledge that Newtonian gravity doesn’t handle light correctly—since it assumes particles with mass and low velocities.

 

[Alien101]

TL;DR Summary: Title: Reframing Light Bending Using Density Instead of Radius

I’ve been exploring an alternative way to express gravitational light bending:

θ = (4G / c²) * (4π / 3)^(1/3) * M^(2/3) * ρ^(1/3)

Instead of using radius r, I substituted it using the density ρ of a spherical mass.

Title: Reframing Light Bending Using Density Instead of Radius

I’ve been exploring an alternative way to express gravitational light bending, starting from the classical general relativity approximation:

θ = (4 * G * M) / (r * c²)

Instead of using radius r, I substituted it using the density ρ of a spherical mass. It should be correct at the radius of a sphere with uniform density.

For a sphere:

ρ = M / V = M / ((4/3) * π * r³)

Solving for r gives:

r = (3M / (4πρ))^(1/3)

Substituting this into the original bending formula gives:

θ = (4G / c²) * (4π / 3)^(1/3) * M^(2/3) * ρ^(1/3)

This reframing expresses light bending as a function of mass and density, rather than radius. It seems to offer a more material-based interpretation of curvature — compactness and mass together determine how much light bends.

The beauty of this is that more mass or more density increase the bending.

I’m curious whether this formulation has been explored before, or if it might offer any new insights into gravitational lensing or curvature fields. Would love to hear your thoughts.

Your formulation suggests that gravitational lensing depends on both the total mass (M^(2/3)) and local density (ρ^(1/3)) with different scaling laws. This implies that a 1000× increase in mass has a different effect than a 1000× increase in density. How do you reconcile this with the equivalence principle, which suggests that only the total enclosed mass within the light ray's closest approach should matter? Shouldn't two objects with identical M but different density distributions produce identical lensing at the same impact parameter?

 

[Ibix]

Your formulation suggests that gravitational lensing depends on both the total mass (M^(2/3)) and local density (ρ^(1/3)) with different scaling laws. This implies that a 1000× increase in mass has a different effect than a 1000× increase in density. How do you reconcile this with the equivalence principle, which suggests that only the total enclosed mass within the light ray's closest approach should matter? Shouldn't two objects with identical M but different density distributions produce identical lensing at the same impact parameter?

You seem to be missing that he's only discussing grazing orbits, so his impact parameter varies with density if the total mass is held constant. That's why his result is only out by constant factors that follow from him using the wrong approxmation.