# Fast as Light

*Most recent answer: 10/22/2007*

Q:

if not possible to travel the speed of light since the weight of energy is more than it is able to move.then how could light move so fast with out having the weight of energy contributing to its speed.

- doug sluga (age 12)

- doug sluga (age 12)

A:

You’re right. Something can not accelerate to the speed of light
without unlimited energy. However, light gets around this somehow. If
light had mass, then we would be in big trouble because it would not be
able to get to the speed of light (but then the speed of light wouldn’t
really be the speed of light). Before we try to think too much about
this, we can avoid the problem by realizing that light has no mass.
This may sound pretty odd but it is true.

Adam

Adam

*(published on 10/22/2007)*

## Follow-Up #1: Energy and mass are different

Q:

You said on one of your answers:
"You’re right. Something can not accelerate to the speed of light without unlimited energy. However, light gets around this somehow. If light had mass, then we would be in big trouble because it would not be able to get to the speed of light (but then the speed of light wouldn’t really be the speed of light). Before we try to think too much about this, we can avoid the problem by realizing that light has no mass. This may sound pretty odd but it is true."
But... energy is mass, is it not? It’s difficult to move something that has, for example, gravitational potential energy, just like it’s difficult to move something that has gravitational potential energy.
Light, we know, is acted upon by gravity. This means it has gravitational potential energy. This kind of energy is difficult to move, as said, which means light should be, also.
Is the answer to my ’question’ that, although light has a sort of mass from energy, that mass does nto become infinite when it moves at C because that mass is not really mass? That is to say: does the gamma function not act upon things like gravitational potential energy?

- Pat van Nieuwenhuizen (age 16)

Ward Melville High School, NY

- Pat van Nieuwenhuizen (age 16)

Ward Melville High School, NY

A:

Hi Pat,

Good question! This is an example of the kind of question that arises when we use E=mc^2 without looking at how it was derived and what restrictions there are.

E=mc^2 describes the energy of a massive particle when it is at rest. Sometimes we call this E the rest energy, and m the rest mass, sometimes writing it as m_0. If a massive particle is moving, its energy increases with its speed. Some people like E=mc^2 so much that they define "m" so that it, too, depends on the speed. I don’t like this because it removes one of two useful concepts -- energy and mass, by equating them. It turns out that the rest mass m_0 is a useful property of objects, even when they are moving at very high speeds, and a slightly more complicated version of Einstein’s famous equation is much more useful in practice:

E^2 = (p*c)^2 + (m_0*c^2)^2

where p is the momentum of the object, in appropriate units I write m_0 to make the old-style people happy, but people who use relativity every day just call it "m", and use the name "invariant mass" because it doesn’t depend on the motion. This relation is true for a collection of particles, too, not just single particles.

For photons, m_0 = 0. This means that E=p*c, and photons have both energy and momentum, but not mass ("rest mass"). We have an answer somewhere on this site which explains why massless objects must travel at the speed of light.

One of Albert Einstein’s great achievements was the reconciliation of special relativity (which is required if the laws of electricity and magnetism is to remain consistent in all frames of reference) and the observation of things which obey Newton’s law of gravitation. General relativity is the outcome of that process, and it’s pretty wild. Perhaps the best way to talk about light being "bent" by a gravitational field, is the attitude taken by Misner, Thorne and Wheeler in their fine book "Gravitation". Light always travels in straight lines. What gravity does is it bends space around so that straight lines are the curved paths we observe light to follow, in a set of coordinates we find convenient.

You can do a rough calculation of the bending of light in a gravitational field by using E=mc^2 for light and treating everything as if relativity otherwise didn’t exist. But the answer you get is a factor of two off.

But the Newtonian picture, which includes the idea of gravitaitonal potential energy, must remain a good explanation of phenomena, at least in its domain of applicability. If you shine lots of light from the surface of the earth to an object way up high on top of a tower, you increase the energy of the object on the tower, and hence its mass (provided its momentum remains zero). Drop this object down to the surface, and its change in potential energy, and hence its kinetic energy, depends on its mass, and therefore on how hot it is, and so somehow that light must have contributed to the gravitational potential energy somehow. To make the relativitstic description work out consistently with this Newtonian picture, light must lose some energy as it goes up in a gravitational field. But its energy is proportional to its frequency, so the frequency of a light ray must decrease as it goes up. There’s a real problem here, since an observer on the ground watching a train of light waves go past counting each crest in a unit of time may compare his answer with another similar observer way up high, watching the same light beam go past (but with a lower frequency). In the same amount of time, the higher observer must count fewer crests of the light wave. Where do the extra crests go?

It turns out you cannot make wave crests go away. But you can explain that the clocks used by the ground-based observer run at a different speed from the observer up high. So the claim that a photon gains potential energy as it loses kinetic energy as it travels up out of a gravitational field is in a Newtonian language which isn’t a full explanation of what’s going on. What’s going on is that the light is traveling from one place to another where the geometry of space and time is different, and the clocks run at different speeds.

Tom

Good question! This is an example of the kind of question that arises when we use E=mc^2 without looking at how it was derived and what restrictions there are.

E=mc^2 describes the energy of a massive particle when it is at rest. Sometimes we call this E the rest energy, and m the rest mass, sometimes writing it as m_0. If a massive particle is moving, its energy increases with its speed. Some people like E=mc^2 so much that they define "m" so that it, too, depends on the speed. I don’t like this because it removes one of two useful concepts -- energy and mass, by equating them. It turns out that the rest mass m_0 is a useful property of objects, even when they are moving at very high speeds, and a slightly more complicated version of Einstein’s famous equation is much more useful in practice:

E^2 = (p*c)^2 + (m_0*c^2)^2

where p is the momentum of the object, in appropriate units I write m_0 to make the old-style people happy, but people who use relativity every day just call it "m", and use the name "invariant mass" because it doesn’t depend on the motion. This relation is true for a collection of particles, too, not just single particles.

For photons, m_0 = 0. This means that E=p*c, and photons have both energy and momentum, but not mass ("rest mass"). We have an answer somewhere on this site which explains why massless objects must travel at the speed of light.

One of Albert Einstein’s great achievements was the reconciliation of special relativity (which is required if the laws of electricity and magnetism is to remain consistent in all frames of reference) and the observation of things which obey Newton’s law of gravitation. General relativity is the outcome of that process, and it’s pretty wild. Perhaps the best way to talk about light being "bent" by a gravitational field, is the attitude taken by Misner, Thorne and Wheeler in their fine book "Gravitation". Light always travels in straight lines. What gravity does is it bends space around so that straight lines are the curved paths we observe light to follow, in a set of coordinates we find convenient.

You can do a rough calculation of the bending of light in a gravitational field by using E=mc^2 for light and treating everything as if relativity otherwise didn’t exist. But the answer you get is a factor of two off.

But the Newtonian picture, which includes the idea of gravitaitonal potential energy, must remain a good explanation of phenomena, at least in its domain of applicability. If you shine lots of light from the surface of the earth to an object way up high on top of a tower, you increase the energy of the object on the tower, and hence its mass (provided its momentum remains zero). Drop this object down to the surface, and its change in potential energy, and hence its kinetic energy, depends on its mass, and therefore on how hot it is, and so somehow that light must have contributed to the gravitational potential energy somehow. To make the relativitstic description work out consistently with this Newtonian picture, light must lose some energy as it goes up in a gravitational field. But its energy is proportional to its frequency, so the frequency of a light ray must decrease as it goes up. There’s a real problem here, since an observer on the ground watching a train of light waves go past counting each crest in a unit of time may compare his answer with another similar observer way up high, watching the same light beam go past (but with a lower frequency). In the same amount of time, the higher observer must count fewer crests of the light wave. Where do the extra crests go?

It turns out you cannot make wave crests go away. But you can explain that the clocks used by the ground-based observer run at a different speed from the observer up high. So the claim that a photon gains potential energy as it loses kinetic energy as it travels up out of a gravitational field is in a Newtonian language which isn’t a full explanation of what’s going on. What’s going on is that the light is traveling from one place to another where the geometry of space and time is different, and the clocks run at different speeds.

Tom

*(published on 10/22/2007)*