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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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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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