Curvature of spacetime, Matter vs Energy

[gc0]

Hello,

I have what is probably a relatively (no pun intended) simple question pertaining to general relativity. I thought that I had achieved a solid understanding of the theory (and its special counterpart), until this question formed in my mind.

My problem is based on the following assumptions:

- Matter and energy cause the bending of space time.
- This bending can affect matter (as evidenced by gravity).
- This bending can also affect energy (as evidence by gravitational lensing).
- Objects with mass in gravitational fields accelerate independent of their mass.

Alright, so here is where the confusion arises:

If the bending of spacetime around a body can cause objects of different masses to accelerate at the same rate (assuming they are at equivalent distances from the body, and there are no sources of friction), then why does it not affect light (photons, energy) equivalently?

Rephrased:

If an atom, a grain of sand, a pea, a cannon ball and a meteor will all accelerate towards the Earth at the same rate regardless of mass or initial velocity (again assuming the same distance and no friction), then why does it take a whole galaxy to alter the course of a photon? If I engage a laser pointer, why does the beam not curve towards the ground at the same time as a bullet fired parallel to the laser? After all, space is curved similarly around both.

I have only a high-school education thus far, so it's entirely possible I'm tackling this issue with the wrong background knowledge.

Your time and knowledge are greatly appreciated. :)

 

[atyy]

A "free" light ray is special in SR and GR - it always follows a null geodesic, whereas ordinary matter follows "time-like" geodesics.

However, "trapped" light in a box does fall like ordinary matter.

This does not answer your question as to "why", but it does say that within the framework of SR and GR there is no answer, it is put in "by hand" in those theories.

 

[gc0]

Thank you for your reply. I understand that in asking 'why' I may very well have been questioning the causes behind something seemingly fundamental (and therefore without meaningful 'cause'). With that said, the different geodesics do offer context for the different behaviors.

 

[HallsofIvy]

I am puzzled by your basic question: "why does it not affect light (photons, energy) equivalently".

It does. The bending of light by a star was one of the first important tests of general relativity. You can't see light bend in the Earth's gravitational field because it is so very fast. If you could throw a ball near the speed of light, you would not see its path bend at all before it was out of sight.

 

[Dale]

Hi gc0, welcome to PF.

I think the premise of your question is wrong. Free particles follow geodesics, the treatment is essentially the same. As atyy mentioned free falling matter follows timelike geodesics and free photons follow lightlike (aka null) geodesics, and as HallsOfIvy mentioned lightlike geodesics near the Earth are also curved. In any given coordinate system there are timelike geodesics that are arbitrarily close to any specific lightlike geodesic.

 

[DrGreg]

The trajectory of a particle depends on both acceleration and "initial velocity". All free-falling particles accelerate in the same way, but massive particles always have an "initial velocity" less than c, and photons always have an "initial velocity" equal to c. That's the difference.

 

[Jonathan Scott]

A "free" light ray is special in SR and GR - it always follows a null geodesic, whereas ordinary matter follows "time-like" geodesics.

However, "trapped" light in a box does fall like ordinary matter.

This is true in a sense, but it is unhelpful because it makes it seem that light behaves differently from matter and differently when "trapped", but neither of these is really the case.

The gravitational field of a central mass has two effects, of equal magnitude: it curves space with respect to displacement in time, and it curves space with respect to displacement in space.

The first of these means that an object which is moving slowly (but is of course moving through time at the usual rate) will accelerate with the usual Newtonian acceleration.

The second of these means that relative to a flat background coordinate system, the curvature of space will cause a horizontally moving object or light beam to experience an additional acceleration equal to v²/c² times the Newtonian acceleration, which adds up to twice the Newtonian acceleration if the object or light beam is traveling at or around c.

Rigid objects such as boxes containing mirrors follow the curvature of space, so are curved relative to a flat background coordinate system. (The bottom and top are curved, and the sides diverge slightly towards the top). This means that relative to the box, the additional effect of the acceleration due to the curvature of space will not be visible, and a light beam will appear to accelerate downwards at the usual Newtonian acceleration.

If instead of acceleration (the rate of change of v), one considers the rate of change of v/c², where c is now the coordinate speed of light relative to the background (isotropic) coordinate system, then the rate of change for an object moving with velocity v is g/c² (1+v²/c²) regardless of the direction of v. This is also proportional to the rate of change of momentum in that background coordinate system, which is given by Ev/c², where E is the energy of the object (which is constant in a static field, as in the Newtonian case where potential energy of a free falling object is converted to or from kinetic energy). This simplification when dealing with the momentum instead of the acceleration can be useful when calculating effects on orbits.

 

[nutgeb]

If an atom, a grain of sand, a pea, a cannon ball and a meteor will all accelerate towards the Earth at the same rate regardless of mass or initial velocity (again assuming the same distance and no friction), then why does it take a whole galaxy to alter the course of a photon? If I engage a laser pointer, why does the beam not curve towards the ground at the same time as a bullet fired parallel to the laser? After all, space is curved similarly around both.

Your question has been answered, but let me try to put it as simply as possible:

Any projectile (or particle) that is initially moving horizontally above the ground (our gravitation source) will trace out a trajectory that curves downward. But the slower the object's horizontal speed, the more sharply its trajectory will curve down. A fast bullet will have a very shallow downward curve, enabling it to travel a long distance before hitting the ground. A stone thrown horizontally will have a sharper downward curve because of its slower speed, and therefore won't travel as far horizontally. At the extreme limit, a rock dropped with no horizontal speed will fall straight to the ground, which is the sharpest possible trajectory curvature.

For a particle whose path does not cause it to fall relatively far vertically through a gravity well, gravity imparts a downward acceleration to a horizontally traveling projectile which is fairly constant as a function of time, not as a function of distance travelled. So the more time that a given projectile spends traveling through a (nearly uniform) gravitational field, the more its trajectory becomes curved downward. In fact, when gravity is constant (and ignoring friction effects), all projectiles fired horizontally, regardless of speed (as long as the speed does not exceed escape velocity), will strike the ground after the same amount of travel time. The projectiles with greater horizontal speeds will travel further horizontally before they strike the ground.

A photon traveling parallel to a horizontal surface has the highest possible speed. Therefore its trajectory will curve down less sharply than the trajectories of much more slowly moving massive objects like cannonballs or the space shuttle.

However, as others said, in General Relativity photons are actually twice as affected by gravity as slow-moving massive objects are. One way to think informally about this is that gravity couples twice as strongly to kinetic energy as it couples to the internal energy of matter, causing twice the total downward acceleration for a photon whose energy is entirely kinetic. (This is intended only as a helpful analogy.)

Despite the extra acceleration, photons travel so much faster than everyday objects (like cannonballs or the space shuttle) that the downward curvature of their trajectory is much, much shallower. The change from a straight-line horizontal trajectory is barely noticeable. And unless the source of gravity is as powerful as a black hole at its event horizon, the photon's speed will always exceed the escape velocity of the gravity, and therefore its initially horizontal path will never be curved downward far enough to cause the photon to hit the ground.

It is not correct to think of spacetime curvature as a fixed physical constraint, like a railroad track, that forces all moving objects to follow exactly the same path. As I've tried to describe, it acts more like a constant influence on moving objects, for example, like a constant wind blowing across the object's path.