How can one define a covariant rate of change of rest mass for an extended body? For a point mass I can define a covariant rate of change of its rest mass as $\frac{d}{d\tau}m_0$ where $\tau$ is the proper time. How can I also define a covariant rate of change of rest mass for an extended body, bearing in mind that different parts will have different world lines and hence proper times?
 A: The covariant rate of change can be obtained by a covariant time derivative operator defined by $$\frac{D}{D\tau}=u^\mu\nabla_\mu$$ where $u^\mu$ is the 4-velocity.
In flat space-time, this definition reduces to $$\frac{d}{d\tau}=u^\mu\partial_\mu$$ 
PROOF:
For an arbitrary scalar $\phi$, we can write $$u^\mu\partial_\mu\phi=\frac{dt}{d\tau}\frac{\partial\phi}{\partial t}+\frac{dx}{d\tau}\frac{\partial\phi}{\partial x}+\frac{dy}{d\tau}\frac{\partial\phi}{\partial y}+\frac{dz}{d\tau}\frac{\partial\phi}{\partial z}\\ \qquad\qquad=\frac{dt}{d\tau}\left(\frac{\partial\phi}{\partial t}+\frac{dx}{dt}\frac{\partial\phi}{\partial x}+\frac{dy}{dt}\frac{\partial\phi}{\partial y}+\frac{dz}{dt}\frac{\partial\phi}{\partial z}\right)$$
The expression inside the brackets can be written as a total derivative using the definition of the material derivative:$$\frac{d}{dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla$$ So, the above equation reduces to  $$u^\mu\partial_\mu\phi=\frac{dt}{d\tau}\frac{d\phi}{dt}=\frac{d\phi}{d\tau}$$
When the scalar $\phi$ is the rest mass, we have
$$\frac{dm_0}{d\tau}=u^\mu\partial_\mu m_0$$
NOTE:


*

*The material derivative takes care of the varying proper time and varying local mass density (if the mass density is not uniform) of the various parts of the extended object.

*The above calculation is valid not only for a scalar, but also for tensors.

