Does \(E_{0}=m c^{2}\) apply only to objects traveling at the speed of light? Why or why not?

At the heart of understanding the universe at high speeds is the theory of Special Relativity, developed by Albert Einstein in 1905. It fundamentally changed our views of space and time, introducing the idea that both are relative rather than absolute concepts. This theory provides a framework for the physics of objects moving at constant speed relative to each other, especially at velocities close to the speed of light.

Special Relativity introduces several mind-bending concepts, such as time dilation and length contraction, suggesting that time can 'slow down' and distances can 'shrink' as objects move faster. One of its core principles is that the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source. This consistency of light's speed leads to the universal maximum speed—the speed of light beyond which nothing can travel.

Rest Energy is a term that emerges from the famous equation, often seen as the most celebrated revelation of Special Relativity: the energy-mass equivalence formula, denoted by \( E_0 = mc^2 \).

The idea is that energy and mass are two sides of the same coin; they can be converted into each other and are, in fact, equivalent. In this equation, \( E_0 \) represents the rest energy, which is the energy inherent in an object solely due to its mass when the object is not in motion, effectively at 'rest'. The rest energy is a form of potential energy, waiting to be converted into other forms, such as kinetic energy or radiation.

Rest energy posits that even objects that are stationary, with no observable movement and no kinetic energy, still contain energy by the virtue of having mass. Thus, an object with mass has an inherent energy, which would be equivalent to the amount of work it could do if it were to be transformed entirely into energy, reinforcing the principle that mass is just a concentrated form of energy.

The speed of light, denoted by \( c \), is a fundamental constant of nature, and its role in physics, particularly in Special Relativity, cannot be overstated. The speed of light in a vacuum is exactly 299,792,458 meters per second. This speed is the cosmic speed limit; in the framework of relativity, no information or matter can travel faster.

In the context of the energy-mass equivalence equation, the symbol \( c \) appears squared, representing an extremely large number. Consequently, even a tiny amount of mass corresponds to a tremendous amount of energy, as seen in nuclear reactions where mass is converted to energy. The speed of light relates not just to the motion of photons, but also serves as a crucial element in the formulas for calculating the energies and momenta of all particles and systems. It's a constant that ensures the same physics applies, whether we are looking at things far out in space or deep within an atom.