# Why do light and masses (like planets) follow different paths through curved space?

It is often said that planets follow a "straight line" through space time. The argument goes that a star like our sun curves space, and the planets follow this path. The argument is also made that light is bent around suns because of this space curvature.

But here's the question: planets and asteroids and space dust will all follow the same orbit (if at the same distance from the sun) regardless of mass. But light doesn't follow the same path. If we explain both light bending and planet orbitals by "the sun curves space," then what is the mechanism by which masses follow a different curved path than that which light follows?

If a massive object would move close to the speed of light and pass at the same distance of the Sun it should describe the same orbital as light.

• So I guess the idea of a "straight path" through spacetime is wrong. If it were truly a straight path, the velocity would not impact the direction. So space is not really "curved" into straight paths that masses and light travel, it just imparts some sort of force that tends to curve the paths of mass and light? – Ralph Berger May 2 '18 at 21:32
• @RalphBerger It's not a straight path through space, it's a straight path through spacetime. Even if two trajectories take the same path through space, if they are at two different velocities, those are two different paths in spacetime. – Chris May 3 '18 at 1:15

The only difference is that masses move slower than light, while light moves at the speed of light (duh). As it turns out, this is a very significant difference. One of the consequences of relativity is that speeds below $c$ are not very different as far as the mathematical formulation goes, because you can move to a different reference frame and from your perspective these speeds will change. The speed of light, however, looks the same to everyone, so light behaves quite differently from not-light.

Mathematically, the four-velocity of a massive object such as a planet (no matter its speed) is a unit vector in four-dimensional spacetime. The four-velocity of light, however, has zero norm (though it is not the zero vector). This puts it into a different category.

Imagine two asteroids, both at the same distance from the sun, both with velocity vectors pointing perpendicular to the distance axis to the sun. If one asteroid has a larger initial velocity that the other, it will follow a different path than the other.

While the trajectories of the asteroids do not depend on their masses, the trajectories certainly do depend on their initial velocities.

There are many "straight lines in space time" that start at the same point and then go in different directions. The difference between these divergent lines is that they start with different initial velocities.

• Thank you, I like this answer. It's not like the sun sets up a single fixed curved space (like in simplistic graphics) where objects come along and follow their assigned path. The sun sets up a curvature "tendency" that objects interact with. The two things - the sun's gravity and the initial velocity of the object - together determine the path, so they together determine the curvature of space. Earth says that the sun has made such a strong curvature, we end up in the same place every 365.25 days. A light beam going by us says that the Sun barely curves space at all. – Ralph Berger May 2 '18 at 21:58
• "It's not like the sun sets up a single fixed curved space ... " What? If you ignore its rotation and treat the Sun as a sphere, the spacetime curvature it creates is described by the Schwarzschild metric. Every particle that comes along, whether it has mass or not, 'sees' the same curvature. – Rodney Dunning May 3 '18 at 0:03
• One of the issues here is the meaning of "straight line" in curved spacetime. This means the shortest possible path which is a geodesic. Objects in free fall (feeling no force) travel on a geodesic, thereby minimizing their proper time. So orbiting around a mass is a "straight line", looking curved though. The main difference talking about asteroids ect. and light is the restmass which is zero for light. Therefore light which has energy and momentum "feels" gravity differently. Note the proper time for a photon is zero. – timm May 3 '18 at 9:00

## protected by Qmechanic♦May 3 '18 at 4:24

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