> OK, color me dense. If photons are massless, how can they be
> perturbed by gravity (one of the tests of Einstein's theories) and
> how can they exert pressure?
Massless gravitational interactions are only a problem in Newtonian
gravity, and even then, Newtonian gravity only leaves the matter
The gravitational interaction between two objects of masses m1 and m2
and separated by a distance r is
F = Gm1m2/r^2
and then the acceleration induced in the object with mass m2 is then
a = F/m2 = Gm1/r^2
Technically, if m2 = 0, as with a photon, then a is undefined. Note,
however, that the discontinuity is removable: We simply let a at m2 = 0
be defined by
a = lim F/m2 = Gm1/r^2
and there is no problem. This trivial extension of Newtonian gravity
(which Newton never considered because he didn't think there was such
a thing as a massless particle) predicts that light passing by the
limb of the Sun is deflected through an angle of about 0.8 arcseconds.
In general relativity, though, gravity is not represented as a normal
force. Instead, it is an effect caused by objects moving through
curved space-time. A massive object like the Sun warps both space
and time; if you move inertially through the warped area, you will
appear to change direction.
Now, an ordinary object moves much more through time than it does
through space: for example, a car moving at 70 mph moves through only
one ten-millionth of a light-second of space for each second of time.
When that car flies off the edge of a cliff (don't try this at home,
kids!), its path is dictated almost exclusively by the warping of
time--almost none of it is due to the warping of space. Newtonian
gravity, in general relativity terms, accounts for the warping of
However, a photon moves through one light-second of space for each
second of time. It is equally affected by both the warp in time
(which Newtonian gravity deals with) and the warp in space (which it
doesn't). Thus, general relativity predicts that the path of an
object in curved space-time depends on its velocity. At one extreme,
a photon's path is deflected by twice the amount predicted by Newtonian
gravity--in the case of the Sun's limb, by 1.7 arcseconds.
This prediction was marginally confirmed by the 1919 eclipse expedition
but has since been better demonstrated in other observations. General
relativity also makes a whole host of other predictions that have been
universally confirmed (to the level of precision of measurement).
As for pressure, photons may have no mass (proper or "rest" mass, that
is), but they do have momentum. Compton demonstrated this when he
showed that light scattered off very light particles was shifted in
frequency, by the proper amount that it would if it were Doppler
shifted because the particles were recoiling.
Since photons have momentum, when they impinge on an object, they must
impart some of that momentum to that object. The rate at which they
impart that momentum is the force (F = dp/dt), and if you divide that
by the area presented by the object, you get the pressure provided by
photons on the object.
Brian Tung <brian@xxxxxxx>
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