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I recently got a more complete proof of photons having no mass. (I knew it before, but now I really know it.) But now, I'm curious how gravitational lensing can occur without a mass to act on.

I have heard that space is like a sheet and gravity works because the more massive an object is, the more it bends space. I heard that when I was five years old. When I got older I questioned how that would work, seeing as space is 3-dimensional. The answer I eventually cobbled together from a plethora of excellent resources was this:

Gravity is like a point light source. At the center, you have the most intense light. As you move outward the intensity decreases with the square of the distance. Like light, gravity radiates in all directions simultaneously.

This works well for me, and I still believe it to be accurate. However, when I was thinking about photons, I realized that you cannot apply a force to an object without mass. At least, you can't by standard Newtonian thinking. This is because $F=ma$. With no mass, you can have no force. Alternately, you could rearrange to $\frac{F}{m}=a$. With no mass, and no force, you can have no acceleration.

Yet gravity is able to refract light.

How is this possible? Like $E=mc^2$, does this only apply to a specific set of conditions?

Qmechanic
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CoilKid
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  • Photons feel gravity. Read about the gravitational red-shift and the Pound-Rebka experiment. – suresh Aug 29 '14 at 00:22
  • I know something about redshift. I was asking if someone could explain how gravity affects photon in general relativity – CoilKid Aug 29 '14 at 00:25
  • @suresh David Hammen mentioned in his answer that in GR, gravity is based on geometry, not force. Could someone please expand on that? – CoilKid Aug 29 '14 at 00:27
  • @suresh also, I meant gravitational lensing as in black holes. We can't see them, but you could see where one was, in theory, by watching for a distortion of the normal background starlight. Like the edges of a glass lens. – CoilKid Aug 29 '14 at 00:30
  • Since you are discovering things for yourself, I did/will not answer your question! The word redshift appears a lot but not all redshifts are the same. So read about the (cool) Pound-Rebka experiment and understand it. – suresh Aug 29 '14 at 00:34
  • Yes, a frequency will redshift as it leaves a large gravitational body. That is partly what I meant. But mostly I'd like to know how a massive object can change the trajectory of a photon that passes at a tangent to the gravitational well. – CoilKid Aug 29 '14 at 00:38
  • @suresh How is it possible for this to happen? – CoilKid Aug 29 '14 at 00:39
  • In a region of uniform gravity, there is no gravity in a freely falling frame(elevator). In such a frame, a photon (and any other particle) in the absence of other forces will undergo uniform motion. If you now go back to the frame where there is uniform gravity, this will give you the motion of a photon in that frame. This has to be next extended to non-uniform gravity and that leads to something called geodesic motion. – suresh Aug 29 '14 at 00:41
  • @suresh

    Photon moving through, for the purposes of this exercise, a region without gravity, will behave similarly to water refraction? like this? *when it moves into a region with gravity

     – CoilKid Aug 29 '14 at 01:05
  • Space is like a sheet and gravity radiates are both wrong as well. (They can be used with the Newtonian gravity in mind, but don't confuse them to the General Relativity.) – firtree Aug 29 '14 at 05:41
  • I was saying that when I was no longer 5, the sheet analogy just didn't cut it. I know gravity doesn't radiate, but it's still a good analogy. – CoilKid Aug 29 '14 at 14:12
  • Possible duplicates: https://physics.stackexchange.com/q/34352/2451 , https://physics.stackexchange.com/q/130552/2451 and links therein. – Qmechanic Dec 14 '19 at 22:47

1 Answers1

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$F=ma$

Don't do that! You can't mix Newtonian mechanics and special relativity, let alone Newtonian mechanics and general relativity.

Gravitation is fundamentally very different between Newtonian mechanics and general relativity. In general relativity, gravitation is a result of geometry. It is not quite a force. Mass-energy tells space-time how to curve. Curved space-time tells mass-energy how to move.

$E=mc^2$

Don't do that, either! A better expression is $E^2 = (mc^2)^2 + p^2c^2$. Note how this allows objects with zero mass to have energy and momentum, objects with zero momentum to have mass and energy. Another way to look at this expression: Energy, mass, and momentum are just different aspects of one fundamental concept. It is this common concept that results in gravitation and interacts with gravitation.


Objects with non-zero mass have energy thanks to that intrinsic mass, making massive objects subject to gravitation. Light has energy thanks to its momentum, so light too is subject to gravitation.

David Hammen
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  • $F=ma$ Yes, I asked if it has special rules like $E=mc^2$. I know that it should be $E^2=(mc^2)^2 + p^2c^2$ In the top of my OP, I said I got a better understanding of why photons have no mass. I meant, I learned that E=mc^2 does not apply to them, and you should instead apply $E^2=(mc^2)^2 + p^2c^2$. – CoilKid Aug 29 '14 at 00:19
  • And Also, I know not to mix Relative and Newtonian physics. I was asking if $F=ma$ had a more applicable version of itself, much like $E=mc^2$ has the Relative physics version of $E^2=(mc^2)^2 + p^2c^2$. That was much the point of my question. – CoilKid Aug 29 '14 at 00:21
  • Thank you for the knowledge that in General Relativity, gravity is based on geometry, not force. Could anyone please expand on that? – CoilKid Aug 29 '14 at 00:23
  • No, for three reasons. Reason #1: You have to learn a lot (a whole lot!) of mathematics to understand general relativity. If you don't have that mathematical understanding you're just fooling yourself. Reason #2: I'm nowhere near as smart as Einstein; I can't reduce GR to a "explain it like I'm five" explanation. Reason #3: While general relativity says what happens with amazing precision, it says nothing about why that stuff happens. General relativity is essentially a kinematic theory. – David Hammen Aug 29 '14 at 00:39
  • Then how the heck can I calculate how much a massive object changes the trajectory of a photon moving at a tangent to the gravity well? I don't need it explained like I'm five. Give me the formula, or a link to somewhere with the formula, and let me figure it out from there! – CoilKid Aug 29 '14 at 00:42
  • Forget formulae. Formulae are definitely not the way to learn physics. Learning physics properly takes a decade. There are no shortcuts. You need to learn Newtonian mechanics and calculus, then you need to learn introductory special relativity, quantum mechanics and partial differential equations. Then you need to learn Lagrangian/Hamilton mechanics and integral equations. Then you need to learn abstract algebra and differential geometry and then maybe you can learn general relativity. And you still haven't tackled group theory and more advanced quantum mechanics. It's a multi-year journey. – David Hammen Aug 29 '14 at 02:11
  • That much I know. I wasn't planning to sit down and work out how much lensing a photon would experience if it passed an object of mass $x$. I was kind of thinking more like, something I could set my desktop wallpaper to, and someday when I do know calculus and quantum mechanics, I could look at it and say "OH. so that's how that works." – CoilKid Aug 29 '14 at 02:17