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Understanding relativity is crucial when analyzing problems involving high speeds comparable to the speed of light. It's the framework developed by Albert Einstein, best known through his Theory of General Relativity and Special Relativity. The latter is especially relevant here as it deals with objects moving at constant high speeds in straight lines, and it introduces the mind-bending concept that time is not absolute but relative—hence 'relativity.' This concept leads to what we call 'time dilation,' where a moving observer measures time to pass more slowly compared to a stationary observer. A practical consequence is that, for high-speed particles in physics experiments, we can observe them lasting longer, from our perspective, than they would if they were at rest. This connects directly to the exercise's scenario, where the \(\Sigma+\) particle's lifetime appears extended, illustrating the warping of spacetime due to its high velocity.

The 'mean lifetime' of a particle, sometimes referred to as 'average lifetime,' is the average time a particle is expected to exist before decaying. This value is a probabilistic measure used in particle physics to describe the stability of subatomic particles. The mean lifetime is a rest frame property—meaning it's the time measured when the particle is not moving relative to the observer. When the particle is moving at speeds close to the speed of light, as observed in the lab frame, the measured lifetime becomes longer due to time dilation, which aligns with the exercise's observations. Time dilation directly affects the mean lifetime calculation, and this change is exactly what allows physicists to infer the speed of high-velocity particles without directly measuring their speed.

Every elementary particle has an intrinsic property called 'rest mass,' sometimes simply called 'mass.' The rest mass is the mass that an object has when it is not in motion relative to an observer. It is a fundamental characteristic of particles and does not change regardless of the speed of the particle. In formulas involving relativity, rest mass is important because it helps us calculate both the rest energy of the particle using Einstein's famous equation \(E = mc^2\), and also its total energy when moving at high speeds. The concept of rest mass is pivotal in understanding how energy and momentum are linked in relativistic physics, as seen when calculating kinetic and total energies in the given exercise.

Kinetic energy is the energy that an object possesses due to its motion. In classical physics, it's simply calculated as \(\frac{1}{2}mv^2\) where \(m\) is the mass, and \(v\) is the velocity of the object. However, this equation becomes insufficient at velocities approaching the speed of light, which is why relativistic physics provides a more comprehensive understanding. In the relativistic setting, kinetic energy is seen as the additional energy a particle gains due to its motion on top of its rest energy. It is derived from the total energy \(E\) minus the rest energy \(E_0 = mc^2\). The exercise provided is a prime example of applying these concepts to a real-world scenario, where the kinetic energy is a critical component in understanding the dynamics of subatomic particles.

Total energy in the context of relativity is the sum of an object's rest energy and its kinetic energy. Mathematically, it is represented as \(E = \gamma mc^2\), where \(\gamma\) is the Lorentz factor, which corrects for the effects of relativity at high speeds, \(m\) is the rest mass, and \(c\) is the speed of light. Unlike classical energy conservation, relativity tells us that mass and energy are interchangeable and so the total energy must account for kinetic, potential, and mass energies. The exercise demonstrates this by requiring the calculation of the total energy in the lab frame, where the particle is in motion, and also in the particle's frame, where the particle is at rest, thus its total energy is simply its rest energy. To understand any particle's behavior at relativistic speeds, grasping the concept of total energy is essential.