Intuitive look at bending of light by gravity

[Owe Kristiansen]

TL;DR

Just realized something elegant by studying ratios:

The bending of light by gravity can be written as:

phi = (2 × potential energy) / (kinetic energy)

It simplifies to phi = 4GM / rc²

That is:
Potential energy: GMm / r
Kinetic energy: ½ mc²

phi = 2 *(GMm / r) / (½ mc²)

This gives the full general relativity result — 1.75 arcseconds for light grazing the Sun — using a simple Newtonian energy ratio.

The factor 2 is needed, not completely sure why.

Mass cancels out, so any mass gives same result.

Comments appreciated.

 

[Ibix]

The factor 2 is needed, not completely sure why

Because you are using Newtonian physics, which is not valid for high velocities like light. It's not exactly a coincidence that the error is exactly a factor of two, but it's not much more than that.

 

[PeroK]

The factor of two is needed because it's possible to predict the bending of light using Newtonian gravity. You can assume light accelerates the same as a massive object.

That calculation gives a different result from the GR calculation. If it didn't, then Eddington's experiment would not have corroborated GR.

 

[Owe Kristiansen]

Because you are using Newtonian physics, which is not valid for high velocities like light. It's not exactly a coincidence that the error is exactly a factor of two, but it's not much more than that.

The standard phi formula is it a newtonian physics formula?

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 appears in the standard phi formula:

4GM / rc²

 

[Ibix]

The standard phi formula is it a newtonian physics formula?

$GM/r$? Yes.

 

[Owe Kristiansen]

$GM/r$? Yes.

I refered to 4GM / rc², that I am using.

 

[Ibix]

I refered to 4GM / rc², that I am using.

That isn't the potential.

 

[Nugatory]

I refered to 4GM / rc², that I am using.

Factors of $c^2$ generally obscure the underlying physics and get in the way of intuitive understanding.

They go away when we set $c=1$ by choosing to measure distances in light-seconds instead of meters (and measuring time in seconds).

 

[Owe Kristiansen]

Yes, that is a smart move. Then speed is one length per one time and so on, replacing meters and seconds. But when checking formulas with real experiments to verify if something is not quite right, measurements are in meters and seconds, not in lightyears per years. :-)

 

[PeroK]

Yes, that is a smart move. Then speed is one length per one time and so on, replacing meters and seconds. But when checking formulas with real experiments to verify if something is not quite right, measurements are in meters and seconds, not in lightyears per years. :-)

In cosmology generally measurements are angles and wavelengths of light. You can't measure anything directly in metres.

if you measure the distance to the Moon by reflecting a light pulse off the surface, then the answer you get is directly in light seconds.

 

[Owe Kristiansen]

One length per time can be anything, unless you anchor it in something. One lightyear can still be converted to meters/seconds for the si units, and also lightyears/year (always 1 in vacum). Lightspeed is also one planc length per planc time, increasibly small units compared to meters per seconds. Still, measuring visible light is useful in nanometers, as this is where it naturally fits. All can be converted to whatever measure you want, as long as there is a valid conversion factor. Ratios is very useful, astronomical units is one, distance from sun to earth. Plancs is not useful here. The math works regardless, if done right. This is about convenience and intuition.

 

[A.T.]

The factor 2 is needed, not completely sure why.

This might help:
https://www.mathpages.com/rr/s8-09/8-09.htm

 

[Dale]

phi = (2 × potential energy) / (kinetic energy)

It simplifies to phi = 4GM / rc²

Kinetic energy: ½ mc²

$\frac{1}{2} mc^2$ isn’t anything. Kinetic energy is $\frac{1}{2} mv^2$ and rest energy is $mc^2$. But nothing is $\frac{1}{2} mc^2$

 

[Hornbein]

I'm sure Albert was glad of that factor of 2 difference. Imagine how disappointing it would have been if the result had been exactly the same as that of Newton.

 

[Owe Kristiansen]

That isn't the potential.

Factors of c2 generally obscure the underlying physics and get in the way of intuitive understanding.

They go away when we set c=1 by choosing to measure distances in light-seconds instead of meters (and measuring time in seconds).

Col

$\frac{1}{2} mc^2$ isn’t anything. Kinetic energy is $\frac{1}{2} mv^2$ and rest energy is $mc^2$. But nothing is $\frac{1}{2} mc^2$

Ok, for a photon, v=c, so I was thinking this was valid. Bending of light happens at light speed

 

[Dale]

Col

Ok, for a photon, v=c, so I was thinking this was valid. Bending of light happens at light speed

For a pulse of light $m=0$.

 

[Owe Kristiansen]

For a photon $m=0$.

You are right. But it still bends in a gravity field. And a photon has both particle and wave properties. So, that it bends could be because both mass and frequency is energy.

In my intuition, this makes some sense, with my limited understanding.

I know that with Einstein its about finding what is the straigt line everything moves along, according to both general and special relativity. I am trying to make sense of all this at the moment.

 

[PeroK]

I am trying to make sense of all this at the moment.

It's time to quote Roger Bacon (1220-1292):

Whoever then has the effrontery to study physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom.

 

[Dale]

But it still bends in a gravity field. And a photon has both particle and wave properties. So, that it bends could be because both mass and frequency is energy.

Sure, but that specific formula isn’t correct. The KE of light is not $\frac{1}{2}mc^2$. For a pulse of light that expression evaluates to zero but light has nonzero KE.

Again, that specific expression is not anything.

 

[phinds]

I know that with Einstein its about finding what is the straigt line everything moves along, according to both general and special relativity. I am trying to make sense of all this at the moment.

That's a great thing to take a look at. The first thing to see is that "straight line" doesn't mean the same geometric construct in Newtonian physics (Euclidean Geometry) as it does in relativity (Reimann Geometry **).

In reality, (that is, the universe we live in) light "bends" only if you try to apply Euclid Geometry in a domain where it does not apply. We SAY, and you will always see this in pop-sci presentation, that light "bends" but that is not strictly correct. In relativity, light, when going around a massive object, goes in a straight line, but that straight line is a Riemannian straight line, called a "geodesic".

Our "intuition" fails us here because we grew up in a world where things move incredibly slowly compared to c, and gravity is too weak for relativity to apply, except out at the 8th decimal place (or so), and Euclidean geometry is correct to an almost immeasurable error bar.

NASA sends men to the moon and calculates orbital trajectories using Newtonian physics. EDIT, I think relativity WAS taken into account for purposes of electronic communication and fine-tuning landing sites.

** "pseudo-Reimann Geometry" to be totally precise

 

[Nugatory]

Ok, for a photon, v=c, so I was thinking this was valid.

The formula that you want is $E^2=(pc)^2+(mc^2)^2$ which works for all particles whether $m=0$ or not.

You are right. But it still bends in a gravity field. And a photon has both particle and wave properties. So, that it bends could be because both mass and frequency is energy.

A photon has both particle and wave properties, but photons have little to do with gravitational deflection of light.

When you hear someone say "photon" in a discussion of relativity, nearly always they should have said "light signal" or "flash of light". As long as you're working with relativity instead of quantum mechanics, you should try to forget that you ever heard the word "photon" - photons aren't what you think they are.

 

[Owe Kristiansen]

The formula that you want is E2=(pc)2+(mc2)2 which works for all particles whether m=0 or not.

Cool formula, I think in terms of energy preservation. Whether it is frequency, mass, potential or kinetic energy. Vibrations, temperature, electric energy and so on. Nature know nothing about straight lines in that sense, and modeling it with coordinate systems that make curved straight is of course ok. A sattelite also moves in a straight line around earth just by newtonian calculations. I am a generalist, so I would like to use the formula that always give the correct result, and if possible understand how the formula gets it right.

 

[Owe Kristiansen]

The formula that you want is $E^2=(pc)^2+(mc^2)^2$ which works for all particles whether $m=0$ or not.

Thanks a lot for a constructive feedback. The formula with E=pc^2 + mc^2 made my day. Its intuitive and replaces my understanding that "photons" / light signals had some kind of mass.

Photons to me is energy packets with light speed and frequency. According to the formula they transport momentum from one place to another then.

 

[Nugatory]

Photons to me is energy packets with light speed and frequency. According to the formula they transport momentum from one place to another then.

I will repeat the advice I gave you in an earlier post: Photons aren't what you think they are.

 

[phinds]

Photons to me is energy packets with light speed and frequency. According to the formula they transport momentum from one place to another then.

You REALLY should listen to @Nugatory

 

[Dale]

@Owe Kristiansen most of the time that you want to use the word “photon” outside of actual quantum mechanics, the term you actually should use is “light pulse”. In particular here, talking about the deflection of rays of light, we are not talking about photons.