which is the ordinary definition of momentum with the mass replaced by the to make use of the Einstein relationship to relate mass and momentum to energy. Relation between momentum and kinetic energy. Note that if a massive particle and a light particle have the same momentum, the light one. you see that you can write the energy in terms of momentum via the light thing will have less momentum, because it will take less time to get.
You're pretty much right. In order for a particle without rest mass to have momentum, it must actually travel at exactly the speed of light. I have a follow up question about "amount" of bending of light in a gravitational field: You mentioned before that "you'd calculate some bending of light in a gravitational field, but it would only be half the observed amount.
For the Newtonian or Special Relativistic calculation, it's easiest to treat the light as a particle, although a wave description would also work.
- Energy momentum Formula
- Does light have mass?
The light is traveling at speed c. Say that the gravitational field g is at right angles to the velocity. That's the part of g that gives curvature.
So long as you make the excellent approximation that the total curvature is very small you can just add the curvatures on the nearly uncurved path. Now what happens in GR? I'm a little more comfortable with a wave picture here.
That part of the curvature we just calculated basically corresponds to the change in kinetic energy when something falls in a field.
Energy is just the same thing as frequency, quantum mechanically.
So what that part consists of is just that the part of the wave closer to the sun is higher frequency, with the wavefronts spaced closer together. Try drawing some wavefronts spaced a little closer together on one side and farther apart on the other, remembering that the wave is propagating at right angles to the fronts. You'll see the wave curve.
Relativistic Mechanical Quantities
Why did I go through that wave exercise? In GR, the gravitational distortion affects not only the time part of spacetime the part we just treated but also the space part.
In a standard coordinate choice, there's extra space, more path length, for the parts of the wave nearer the sun. Projection of that curved space onto a flat plane makes waves which have equal spacing nearer and farther from the sun look like they are more tightly spaced near the sun, since the flat space doesn't have that extra stretch in there.
So that's the source of the extra bending. This second bending is just as big for massive particles as for light, but for particles moving slowly compared to c, the first component of the bending is so much bigger that it's hard to see this second part. Why are the two components of equal importance for light?
Energy momentum Formula
In dividing the effect up into the two parts, we've implicitly assumed a conventional coordinate system in which the speed of light is isotropic and everywhere constant.
It looks like the time-like part of the effect comes mainly from the region where the light gets closest to the sun. The space-like part seems to come mostly from two regions, where the light is approaching and leaving that region of closest approach. The reason for that is that the peculiar lengths in this choice of coordinates are only along the radial direction, not along the tangential direction.
Unfortunately I don't yet have a deep enough feel for the calculation to summarize simply why the two parts have to come out equal. As for the observational results, there are many references in. How can a massless object such as light have a momentum? We could then consistently talk about the light having mass independently of whether or not it is contained.
If relativistic mass is used for all objects, then mass is conserved and the mass of an object is the sum of the masses of its parts. However, modern usage defines mass as the invariant mass of an object mainly because the invariant mass is more useful when doing any kind of calculation. In this case mass is not conserved and the mass of an object is not the sum of the masses of its parts. Thus, the mass of a box of light is more than the mass of the box and the sum of the masses of the photons the latter being zero.
Relativistic mass is equivalent to energy, which is why relativistic mass is not a commonly used term nowadays.
In the modern view "mass" is not equivalent to energy; mass is just that part of the energy of a body which is not kinetic energy. Mass is independent of velocity whereas energy is not. Let's try to phrase this another way. You can interpret it to mean that energy is the same thing as mass except for a conversion factor equal to the square of the speed of light. Then wherever there is mass there is energy and wherever there is energy there is mass. In that case photons have mass, but we call it relativistic mass.
Another way to use Einstein's equation would be to keep mass and energy as separate and use it as an equation which applies when mass is converted to energy or energy is converted to mass--usually in nuclear reactions.Relativistic Energy-Momentum Relation
The mass is then independent of velocity and is closer to the old Newtonian concept. In that case, only the total of energy and mass would be conserved, but it seems better to try to keep the conservation of energy. The interpretation most widely used is a compromise in which mass is invariant and always has energy so that total energy is conserved but kinetic energy and radiation does not have mass.
The distinction is purely a matter of semantic convention. Sometimes people ask "If light has no mass how can it be deflected by the gravity of a star? One answer is that all particles, including photons, move along geodesics in general relativity and the path they follow is independent of their mass. The deflection of starlight by the sun was first measured by Arthur Eddington in The result was consistent with the predictions of general relativity and inconsistent with the newtonian theory.
Another answer is that the light has energy and momentum which couples to gravity. The energy-momentum 4-vector of a particle, rather than its mass, is the gravitational analogue of electric charge. The corresponding analogue of electric current is the energy-momentum stress tensor which appears in the gravitational field equations of general relativity.
The energy and momentum of light also generates curvature of spacetime, so general relativity predicts that light will attract objects gravitationally.
This effect is far too weak to have yet been measured.