# Light has no mass so it also has no energy according to Einstein, but how can sunlight warm the earth without energy?

Category: Physics

Published: April 1, 2014

By: Christopher S. Baird, author of The Top 50 Science Questions with Surprising Answers and physics professor at West Texas A&M University

Light indeed carries energy and accomplishes this without having any mass. The Einstein equation that you are probably referring to is *E* = *mc*^{2}. This equation is actually a special case of the more general equation:

*E*^{2} = *p*^{2}*c*^{2} + *m*^{2}*c*^{4}

In the above equation, *E* is the total energy of the particle, *p* is the momentum of the particle (which is related to its motion), *c* is the speed of light, and *m* is the mass of the particle. This equation can be derived from the relativistic definitions of the energy and momentum of a particle. The above equation tells us that the total energy of a particle is a combination of its mass energy and its momentum energy (which is not necessarily related to its mass). When a particle is at rest (*p* = 0), this general equation reduces down to the familiar *E* = *mc*^{2}. In contrast, for a particle with no mass (*m* = 0), the general equation reduces down to *E* = *pc*. Since photons (particles of light) have no mass, they must obey *E* = *pc* and therefore get all of their energy from their momentum.

Now there is an interesting additional effect contained in the general equation. If a particle has no mass (*m* = 0) *and* is at rest (*p* = 0), then the total energy is zero (*E* = 0). But an object with zero energy and zero mass is nothing at all. Therefore, if an object with no mass is to physically exist, it can never be at rest. Such is the case with light. Furthermore, if the object travels at some speed *v* that is less than the universal speed limit *c*, we can always choose a reference frame traveling along with the object so that the object will be at rest in this reference frame. Therefore, an object that can never be at rest must always travel at the universal speed limit c, because this speed has the interesting property that once an object goes a speed *c* in one reference frame, it goes the speed c in all reference frames. In summary, all objects with no mass can never be at rest and must travel at speed *c* in all reference frames. Light is such an object, and the universal speed limit *c* is named the speed of light in its honor. But light is not the only massless object. Gluons and the hypothetical gravitons are also massless, and therefore travel at speed *c* in all frames.

How can an object have momentum without mass? It can do this if it is a wave. A wave transports momentum via its waving motion and not by physically transporting an object with mass. "Momentum" is the directional property of an object in motion that describes its ability to influence another object upon impact. An object with high momentum (such as a truck) can greatly influence the object it collides with (such as a barrel). If a giant water wave collides with a barrel, it can also influence the barrel to move. The water wave therefore carries momentum even though it has no mass. The water itself has mass, but the wave has no mass. A water wave is not a packet of water traveling along. In fact, the water that the wave is traveling through stays more or less in one place. Rather, the wave is a rippling domino-effect of motion. As another example, consider a long jump rope held taut at both ends by two girls. If one girl shakes her end of the rope violently enough to send a wave down the rope to the other girl, the wave can jerk the other girl. The rope has not transported any mass, but it still carries momentum through its waving motion. In this way, waves can have no mass but still carry momentum. In addition to being a particle, light is also a wave. This allows it to carry momentum, and therefore energy, without having mass.