# Special relativity - Maximum mass

• Aleolomorfo
In summary, the maximum mass that can be produced from a collision of identical particles with mass is when the three resultant particles are at rest in the CM frame.
Aleolomorfo

## Homework Statement

Finding the maximum mass ##M_x## which can be made from a collision of identical particles with mass ##m##, in the laboratory frame, in which one particle is at rest and the other one has energy ##E##. The reaction is the following: ##a+b \rightarrow a+b+x##.

## The Attempt at a Solution

I assume that the maximum mass is produced when the three resultant particles are at rest in the CM frame. So I use the invariant of the total momentum squared from the lab frame before the collision (##p##) and the CM frame after the collision (##k##):
$$p_1=(E,0,0,\sqrt{E^2-m^2}) \hspace{1cm} p_2=(m,0,0,0)$$
$$(p_1+p_2)^2 = 2m(E+m)$$
$$k_1=k_2=(m,0,0,0) \hspace{1cm} k_x=(M_x,0,0,0)$$
$$(k_1+k_2+k_x)^2=4m^2+4mM_x+m^2_x$$
I equal the invariants:
$$2m(E+m)=4m^2+4mM_x+M^2_x$$
After calculation I find a second order equation in ##M_x## whose solutions are:
$$-2m\pm\sqrt{2m^2+2mE}$$
I reject the solution with - because it is completely negative but also the solution with + is negative under a certain value of E. I do not know if this is because a mistake or it is the threshold energy of the reaction.

Aleolomorfo said:

## Homework Statement

Finding the maximum mass ##M_x## which can be made from a collision of identical particles with mass ##m##, in the laboratory frame, in which one particle is at rest and the other one has energy ##E##. The reaction is the following: ##a+b \rightarrow a+b+x##.

## The Attempt at a Solution

I assume that the maximum mass is produced when the three resultant particles are at rest in the CM frame. So I use the invariant of the total momentum squared from the lab frame before the collision (##p##) and the CM frame after the collision (##k##):
$$p_1=(E,0,0,\sqrt{E^2-m^2}) \hspace{1cm} p_2=(m,0,0,0)$$
$$(p_1+p_2)^2 = 2m(E+m)$$
$$k_1=k_2=(m,0,0,0) \hspace{1cm} k_x=(M_x,0,0,0)$$
$$(k_1+k_2+k_x)^2=4m^2+4mM_x+m^2_x$$
I equal the invariants:
$$2m(E+m)=4m^2+4mM_x+M^2_x$$
After calculation I find a second order equation in ##M_x## whose solutions are:
$$-2m\pm\sqrt{2m^2+2mE}$$
I reject the solution with - because it is completely negative but also the solution with + is negative under a certain value of E. I do not know if this is because a mistake or it is the threshold energy of the reaction.
What values of ##E## are you worried about?

Aleolomorfo said:
After calculation I find a second order equation in ##M_x## whose solutions are:
$$-2m\pm\sqrt{2m^2+2mE}$$
I reject the solution with - because it is completely negative but also the solution with + is negative under a certain value of E. I do not know if this is because a mistake or it is the threshold energy of the reaction.
Remember ##E## is the energy of a particle of mass ##m##.

Yes, I get it. I've realized the stupidity of my question. ##M_x## is negative if ##E < m##, but this is impossible. Thank you.

Douglas Sunday and PeroK

## 1. What is the theory of special relativity?

The theory of special relativity is a fundamental principle in physics that describes the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, and the speed of light in a vacuum is constant for all observers.

## 2. What is the maximum mass that can be reached according to special relativity?

According to special relativity, the maximum mass that can be reached is infinite. This means that as an object approaches the speed of light, its mass will increase without limit. However, in practical terms, this is not possible to achieve.

## 3. How does special relativity affect the concept of mass?

Special relativity states that mass and energy are equivalent, and they can be converted into each other. This means that the mass of an object is not constant and will increase as it approaches the speed of light. This concept is known as mass-energy equivalence, as described by Einstein's famous equation, E=mc².

## 4. Is there a limit to the speed of light according to special relativity?

Yes, according to special relativity, the speed of light is the maximum speed that anything can travel in the universe. This speed is approximately 299,792,458 meters per second in a vacuum. Nothing can travel faster than this speed, and as an object approaches it, its mass and energy increase without limit.

## 5. How does special relativity impact our understanding of time and space?

Special relativity states that time and space are relative concepts and are intertwined. The theory suggests that time can slow down or speed up depending on an observer's relative speed and the strength of the gravitational field they are in. This also affects our perception of space, as distances can appear shorter or longer depending on the observer's frame of reference.

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