# Why are massive bodies following a different trajectory in a gravity field than light?

I often read, that light is being bent in a gravity field and so are the paths of massive bodies (e.g. planets or stars). I also read that the curvature of space (caused by some other mass) is the reason for this. So why are planets following different paths than light. E.g. Why I don't see light orbiting the sun.

• Light absolutely can form closed orbits en.wikipedia.org/wiki/Photon_sphere – Triatticus Dec 29 '19 at 15:53
• @Triatticus -- yes I know and this was not my question. I already know, that light can be bent by gravity. – Frank Dec 29 '19 at 15:54
• The inportant thing is particles ans light follow the curvature of space time and not just the curvature of space – lalala Dec 29 '19 at 17:22
• Your question is about curved spacetime, and you got some good answers. But it is worth pointing out that the question "why do I not see light orbiting the sun" is self-contradictory. If you can see light then it is inside your eyes, and therefore not orbiting the sun. – Eric Lippert Dec 30 '19 at 9:05
• @EricLippert Good nitpick, although "see" can also mean "experience", and the question can be understood as "Why don't we experience light orbiting the sun". We can also "see" black holes even though they don't emit any light, because we detect their effect on other material nearby. – Barmar Dec 30 '19 at 16:59

Light and matter both follow the curvature of spacetime when passing a massive object. The difference is that matter is ALWAYS slower than light, it will be in the more curved spacetime longer, so it's path is curved more than light. Compare it to throwing a ball, if you throw the ball very slowly, it will follow a sharp curve and fall down close to you. If you throw the ball as fast as you can, it will follow a much longer curve before landing. So you see that the speed changes the trajectory.

• I love these kinds of answers that answer the question succinctly and with accessible language and metaphors – Neil G Dec 30 '19 at 22:02

The relativistic way of looking at things is that objects don't have paths through space, they have paths through spacetime. These are called world-lines. Even in Galilean physics, the trajectory of a test particle through space is not really well-defined. For example, the planet earth doesn't have a well-defined trajectory through space that is an ellipse; it depends on what frame of reference you use. The ill-defined nature of trajectories through space gets even worse in relativity.

For test particles, these world-lines are a type of curve called a geodesic, which is defined to be straight. A ray of light always has a world-line that is "light-like," meaning that it moves at $$c$$. A material object always has a world-line that is "time-like," meaning that it moves at $$.

So both types of test particles follow geodesic paths (which are by definition straight) through spacetime, but they're geodesics that are in two completely disjoint categories.

• @safesphere In standard usage, two sets are disjoint if they don't intersect. The timelike geodesic of a body traveling with uniform speed arbitrarily close to $c$ in some frame can be transformed to a geodesic with zero speed in another frame; you simply can't do that with light-like geodesics. – PM 2Ring Dec 30 '19 at 12:41
• I think that focussing on the distinction between light-like and space-like world lines is a distraction here: the difference in the paths taken by a photon and a planet are due to their different speeds in the frame of the observer. v=c is a special case for many reasons, but it doesn't really change the storyline here. – CharlieB Dec 30 '19 at 20:18

The orbital trajectory taken is dependent on the velocity of the body. Since light always travels at $$c$$ and nothing with mass can, the orbits will always be different. Light does curve around the sun in a hyperbolic path. If there was something with a much more powerful gravitational field than the sun (like a black hole), light would orbit around it. This is the reason there is a ring of light just outside the event horizon of a black hole.

• so to claim, that light (or a massive body) follows the curvature of space is plain wrong? – Frank Dec 29 '19 at 15:59
• so the photon experiences a different curvature than a massive body? – Frank Dec 29 '19 at 16:01
• It experiences the exact same curvature, but due to its extremely high speed the path taken comes out to be different. Similar to how you can take sharper turns when you drive your car slowly but when you increase the speed, the turning radius becomes larger. – Sam Dec 29 '19 at 16:04

What general relativity describes as curved is the space-time continuum. Objects travel through this continuum on geodesics. Like a line in flat 3D space, geodesics have a direction. You don't expect two lines with a common point but different directions to continue together, do you?

Same with the world lines of your particles: The direction of the geodesic that a particle travels on depends on the speed of the particle, and as such the direction of a photon's geodesic must be different from that of any particle that moves at some lower speed.

Formally, the 4-velocity of any particle (at rest, moving, light like) is a vector that is normalized: It always has the same length. For an object at rest, the 4-velocity point only in the direction of time, for a photon it points only in a spatial direction. The faster the particle, the less the 4-velocity point into the time direction, and the more it points into a spatial directions. Consequently, the particle follows a different geodesic depending on its speed.