Question regarding the general form of four-vectors in particle physics

Why is there a $m_X c$ in the first element of a four-vector which (I thought) should have dimensions of energy, not momentum?

The momentum 4-vector of a particle of mass $m$ and three-velocity $mathbf{u}$ can be defined as:

$$P = (gamma m c, gamma m mathbf{u}) = left(frac{E}{c}, mathbf{p}right) tag{A}$$ or as

$$ P = (gamma m c^2, gamma m cmathbf{u}) = (E, mathbf{p}c) tag{B}$$

Note that convention B is just A multiplied by $c$; all components in A have dimensions of momentum, whereas all components in B have dimensions of energy.

Also, I've entered 3-vectors into these expressions as a shorthand to represent the 3 associated components, e.g. A should really be written:

$$P = (gamma m c, gamma m mathbf{u}) = (frac{E}{c}, p_x, p_y, p_z)$$

In the solution, they've used convention A, but you would get the same answer if you used convention B instead - it's just that all your equations would be multiplied by $c^2$ now. I suggest that you check this directly!

Instead, let's do a different check to see if we recover the $E^2 = |mathbf{p}|^2 c^2 + m^2 c^4$ relation using the invariance of the momentum four-vector norm for both cases. We're going to work out the norm squared explicitly in both cases in the rest frame of a particle of mass $m$ (so $gamma = 1$ and $mathbf{u} = 0$) and then equate them to a general expression in terms of $E$ and $mathbf{p}$.

Case A: $$m^2c^2 = frac{E^2}{c^2} - |mathbf{p}|^2 tag{1}$$ Case B: $$m^2 c^4 = E^2 - |mathbf{p}|^2 c^2 tag{2}$$

We see that equation 2 is just equation 1 multiplied by $c^2$ and that both are correct. This makes sense as convention B has just scaled the four-vector in convention A by a factor of $c$ - any equation involving the same relations between these four-vectors should then appropriately be scaled overall by some power of $c$ (which has no effect overall). The solution produces an equation involving two-fold products of these four-vectors, so using convention B will scale the equations by $c^2$.

To answer the final part of this question, you have an $m_X c$ because $E_X = m_X c^2$ in the rest frame of X.

It was my understanding that general four-vectors are written as $(E, p_x c, p_y c, p_z c)$. I thought that elements of four vectors must all have the same dimensions and that those dimensions are energy (as above).

I believe the first part of this question has now been addressed. However, I would like to add that the momentum four-vector (whichever convention you choose) is not the only four-vector that exists.

Actually, $X = (ct, x, y,z)$ is probably the most basic four-vector; this transforms as $X' = Lambda X$ under Lorentz transformations (where X' labels the transformed components of the same four-vector in the new inertial frame and $Lambda$ is the Lorentz transformation matrix).

Now, any 4-component object $A$ that transforms like $A' = Lambda A$ when $X$ transforms like $X' = Lambda X$ is a 4-vector. The elements of $A$ should all have the same dimensions, but they don't have to be energy.

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$$P_X=left(E_X,,boldsymbol{p_X}cright)=(m_Xc^2,0,0,0)$$ $$P_c=left(E_c,,boldsymbol{p_c}cright)=(E_c,{p_c}^xc,0,0)$$ So $$P_Xcdot P_c=E_cm_Xc^2-0=E_cm_Xc^2 tag{C}$$ substituting this result in (1): $$m_d^2c^2=m_X^2c^2+m_c^2c^2-2E_cm_Xc^2 tag{D}$$ $$implies E_cstackrel{color{red}{{?}}}{=}frac{m_X^2+m_c^2-m_d^2}{2m_X}$$ Well, this is definitely not the same answer as (A). So what am I missing?

Step C is correct, but step D is incorrect: under convention B, the norm squared of the momentum four-vector is also scaled by $c^2$. This is because in the rest frame of a particle of mass $m$, $P = (mc^2, mathbf{0})$ so the norm-squared is now $m^2 c^4$ (unlike $m^2 c^2$ under convention A).

If you use this norm, you will recover the correct answer - let's check:

$$P_d^2=P_X^2+P_c^2-2P_X cdot P_c tag{E}$$ is a relation between four-vectors and holds under both conventions.

Using the corrected norms and your expression for $P_X cdot P_c$ , we find:

$$m_d^2 c^4 = m_X^2 c^4 + m_c^2 c^4 - 2 E_c m_X c^2,$$ which is just

$$m_d^2c^2=m_X^2c^2+m_c^2c^2-2 E_c m_X$$ scaled by $c^2$ as expected.

Conventionally, the four-momentum is defined as $$P=(E/c,p_x,p_y,p_z)$$ so that all components have units of momentum. If you want to get the energy of a particle (in units of energy), you need to multiply the zero component of the four-momentum by $c$. This is one example of the central role the speed of light plays in relativity. It allows us to relate energy and momentum. Before it was realized that the speed of light was such an important fundamental quantity, there really wasn't a way to combine energy and momentum into a single concept. In classical mechanics, dimensional analysis would disallow such a relationship.

Edit: $P$ is defined to use units of momentum because this allows for a natural generalization from classical momentum. In classical mechanics, $mathbf{p}=mmathbf{v}$. In relativity, we write $P=mu$, where m is mass and $u$ is the four-velocity of the particle. The four-velocity combines the notion of movement through space with the notion of movement through time. We say that a particle in its rest frame travels only through time (and not through space). Its four-velocity is defined as $u=(c,0,0,0)$. The reason the speed of light enters here is because it allows us to restrict the maximum allowed velocity of the particle to be the speed of light. Any boost preserves $u^2=c^2$. Therefore, in a boosted frame, where $u=(v_t,v_x,v_y,v_z)$, we will still have $u^2equiv v_t^2-v_x^2-v_y^2-v_z^2=c^2$, which implies that the velocity of a particle in a boosted reference frame will never be measured as being faster than light. From this definition for $u$, we get that $P^2equiv m^2u^2=m^2c^2$ in the rest frame, which also means that $P^2=m^2c^2$ in any reference frame.