In
this section we will now pursue the goal of understanding how the field
equation of Einstein is obtained.

The
Einstein’s field equation consists of two crucial concepts, one of them being
the geometry of the curved spacetime and the other being the cause of this
curved spacetime. Einstein believed that this curvature of spacetime was due to
the presence of mass or in other words, “The effect of the mass-energy creates
a curvature in the spacetime.”

We
have discussed the geometric part (curved spacetime) of the equation in the
above mathematical section; however we have not discussed the idea of the
momentum-energy tensor (which is the source of the curvature).

While
we may be tempting to answer the question about why mass effects spacetime, we
have to understand that, it isn’t an easy problem. However, this does not stop
us from assuming that mass does curve the spacetime. We in fact use the
assumption that the curvature of spacetime reduces to the Newtonian approximation
in the limits of the weak field and low density without any known mechanism for
the curvature itself.

What
then is an Energy-Momentum tensor?

Let
us consider a four-vector $p^{\alpha}$. The components of
this four-vector momentum are the energy and the three momentum vectors. The
question that is important here is, how can we associate four-vector $\Delta p^{\alpha}$
with the three volume $n_{\beta} \Delta V?$. The answer for this question is a
tensor of an object with two indices $T^{\alpha\beta}$. This object is called
the energy-momentum Tensor. We can write down the relationship as follows,

To understand what the components of
the energy-momentum tensor are, consider a particular inertial frame in flat
spacetime and a three-dimensional volume $\Delta$ at rest in that frame. That
volume is part of a

$(2)$

Now from $(2)$ it follows that,

*t*= constant. We can choose the normal of the three-surface in spacetime to be $n_{\alpha}=(1,0,0,0)$. With this choice of normal equation $(1)$ becomes,$(2)$

Now from $(2)$ it follows that,

&

Thus we understand the significance of four of the components of the stress-energy tensor. Similarly if we consider a time like three volume spanned by intervals $\Delta y, \Delta z, \Delta t$ then, the unit normal to this three surface pointing in the x-direction is $n_{\alpha}=(0,1,0,0)$. The analog is then,

Thus we understand the significance of four of the components of the stress-energy tensor. Similarly if we consider a time like three volume spanned by intervals $\Delta y, \Delta z, \Delta t$ then, the unit normal to this three surface pointing in the x-direction is $n_{\alpha}=(0,1,0,0)$. The analog is then,

Now this
can be written in general force term as,

Thus,
$T^{ij}$= $i^{th}$ component of the force per unit
area exerted across a surface with normal in direction $j$

In
classical mechanics a force per unit area is called a stress, and $T^{ij}$
is called the stress tensor. In special case when $(i=j)$, we have the usual
pressure components. All in all we can summarize the $T^{\alpha\beta}$ as the
following,

Where, $\varepsilon ,\pi
^{i},T^{\alpha\beta}$ are the energy density, momentum density and the stress
tensor respectively.

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