**4.**

**The Principle of Equivalence and Geodesics**

Up to now, all the tensor descriptions have been
for 3-space and we’ve not introduced time. But this deficiency must now be
remedied as we provide for an extension and generalization of the Principle of
Equivalence to incorporate the fact that the velocity of light is the same in
all reference frames.

To put
it into terms consistent with the examples seen in Part 1, a mass point carried by a gravitational field does not remain at
rest but “falls down”. More to the
point, the path taken for this freely falling mass is a geodesic in space-time
or a “world line” for the mass, i.e. described in 4 dimensions.

Thus,
the formalism or representation must include changes in time and so the
gravitational field is now given as a

**space-time tensor**. Unlike the tensors shown in the previous section, this one displays 4 x 4 components, as opposed to 3 x 3.The reframed Principle of Equivalence therefore states that the equations of motion for this freely falling particle are expressed:

**d**

^{2}x^{k}/ds^{2}+ {mn, k} dx^{m}/ds dx^{n}/ds = 0(where m,n, k = 1,2,3,4 for the four dimensions which are then displayed as: x

_{1}, x

_{2}, x

_{3}and x

_{4}– the last instead of t). It is important to point out that s, the interval, is assigned the dimension of time and hence is the

*proper time*. This is the time measured by a clock traveling with the mass particle. By virtue of the definition and the above equation, the observer has a full definition of a curve traced out in

**space-time**by a moving clock – which possesses an invariant form – and thus has a coordinate system for all events in the universe.

Let a light ray now follow this path and one
obtains a null geodesic, i.e.

Let the upper end of the curve be defined by event of form

**t, x**and the lower extremity by event of form,_{r}**t + dt, x**_{r}**+ d****x**Then for a clock traveling_{r }*with the mass*the proper time between the two events is zero – given the path is a null geodesic. Hence:**g**

_{44}(t, x_{r}) dt^{2 }= 1/ c^{2}**å**

^{3}

_{l,m = 1}g_{lm}**(t, x**

_{r}) dx^{ℓ}**dx**

^{m}
However, for a clock

*at rest*at position x_{1}, x_{2}, x_{3}the*proper time*elapsed between the events shown in the diagram is:

**ds**

_{o}= [g_{44}(t, x_{r})]^{1/2}dt

The
distance between the two events exclusively in space coordinates is:

d
ℓ = å

^{3}_{l,m = 1}g_{lm}(t, x_{r}) dx^{ℓ}dx^{m}^{}^{}

Then,
taking the derivative: d ℓ/ ds

_{o }_{}_{}

**d ℓ /**

**ds**

_{o }=__+__c

where c is called

**at the event (t, x***the velocity of light*_{r}). Hence, we see it emerges as a universal constant and always takes the least time between two events, hence the path taken of a null geodesic. This is consistent with the Einsteinian postulate – also given in special relativity – that the velocity of light*has the same value at all points in space –time.*
Thus we see that in order to include non-uniform
fields the Principle of Equivalence can be reformulated – to a “strong”
form, to read[1]:

“

*In a small, freely falling laboratory, the laws of physics are the same as the laws of special relativity without any gravitational field*.”**5. Einstein’s Field Equations**

In the previous section we saw the
coefficient

**g**appear for the expressions to do with proper time. But what is it? This denotes the gravitational potential field given by a particular set of components, usually written as_{44 }**g**_{m}**for the tensor g. We can now pose the interval introduced in Sec. 4. as:**_{n}**ds**

^{2 }= g

_{m}

_{n}**dx**

^{m}**dx**

^{n}
Then the

**g**denotes the 16_{44 }^{th}and last element of the**g**_{m}**matrix. In flat space-time the tensor components for g are usually given as:**_{n}**(**

**g**

_{11}

**g**

_{12}

**g**

_{13}

**g**

_{14}**)**

(

(

**g**

_{21}**-**

**g**

(g

(g

_{22}g_{23}g_{24})(g

_{31}g_{32}g_{33}g_{34})(g

_{41 }g_{42}g_{43}g_{44})
For
which it is convenient to specify special values of the potentials presented in
“standard form” as:

**g**

g

g

_{11}g_{12}g_{13}g_{14}_{ }g_{22}g_{23}g_{24}g

_{33}g_{34}g

_{44}
Now, for

**flat space-time**the values are all 0 except for those long the diagonal for which:**g**

_{11}**= -1**

**g**= -1

_{22}**g**

_{33}**= -1**

**g**

_{44 = 1}
Thus, we see that the value of

**g**in the previous expressions is 1. Writing out the interval form for the above is straightforward and one only needs to include the correct subscripts for the respective dx’s, in each component, e.g._{44 }**g**

_{11}dx

_{11}

**and**

^{ }

^{ }**g**

_{12}dx

_{1}**dx**

_{12}
Now,
if we take:

**¶****g**_{m}**/**_{n}**¶****x**_{t}
We obtain the

*Riemann-Christofel*curvature tensor:

**G**

^{t}

_{ }

_{m}

_{n}

_{ }**= G**

_{m}

_{n}
For flat space-time the gravitational potentials
satisfy:

**G**

_{m}

_{n}**=0**

Where the values conform to those of the g’s
shown above and we find:

**g**

_{11}

**= -1/**

**g**

**g**

_{22}**= -x**

_{1}^{2}**g**

_{33}**= -**

**x**

_{1}^{2}**sin**

^{2 }**x**

_{2}^{2}

**g**

_{44 }

_{= }**g**

Where:

**g****= 1 -****k**/**x**_{1}
And the constant

**k**is what’s called the Gaussian curvature. It can assume values of 0 (Euclidean 4-D flat space), -1 (Lobachevskian space) or +1 (Riemannian space).
The Einstein field equations can be summarized
in the tensor form:

**G**

_{m}

_{n}**= -**

**½ g**

_{m}

_{n}

_{ }**G= - 8**

**p**

**T**

_{m}

_{n}

Where the

**T**_{m}**denotes the associated “stress-energy” tensor which incorporates internal stresses, the density of matter and its component velocities (u, v, w or in some texts: u1, u2 and u3). From this one can see that if no matter is present, one would have:**_{n}**G**_{m}_{n}**= 0**

If matter is present there must then be internal
stresses and velocities so that:

**G**_{m}_{n}**= K**_{m}**where (as seen from the field equations):**_{n}**K**_{m}_{n}**=****- 8****p****T**_{m}_{n}
The introduction of

**K**_{m}_{n}**to describe the matter-associated properties is generally attributed to Arthur Eddington[2].**
We
have then for the

**T**: analogous to the g’s in standard form_{mn}**T**

T

T

_{11}T_{12}T_{13}T_{14}_{ }T_{22}T_{23}T_{24}T

_{33}T_{34}T

_{44}

_{}**=**

**p**

_{11}+**r u**

^{2}**,**

**p**

_{12}+ uv, p_{13}+**r uw, -**

**ru**

_{ }p_{22}+**r v**

^{2}**, p**

_{23}+**r vw**

**, -**

**rv**

p

p

_{33 }+**r w**

^{2}**, -**

**rw**

**r**

Which again, is a vastly
simplified presentation. It should also be said that radiation via an electromagnetic energy tensor can also be
included – given these will have the same rank as the T

_{m}**These components can thus be added to components of the mass energy tensor shown. In other words, the presence of radiation is taken to be equivalent to the presence of mass, given m = E/c**_{n}^{2}.
One final point: The Riemann curvature
tensor is often written:

**R**

^{a}

_{b}

_{ }

_{m}

_{ }

_{n}And so it serves as a quantitative measure of the curvature in space-time. In flat space-time then, R

^{a}

_{b}

_{ }

_{m}

_{ }

_{n}= 0. Conversely, when R

^{a}

_{b}

_{ }

_{m}

_{ }

_{n}¹ 0 matter is present and space-time is curved. Thus,

**g**

_{m}

_{n}**is a metric tensor from which the Riemannian curvature tensor can be calculated[3].**

_{ }**g**

_{m}

_{n}**has a similar role to the vector potential**

_{ }**A**of electrodynamics. The curvature tensor plays a similar role to the

**E, B**fields. Thus we know

**Ñ X A**=

**B**, for example.

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