The study of motion of the object is an important section of the physics. The motion of a body is can be measured as absolute motion and relative motion. Practically any motion is measured is relative only, because one or the other way all the bodies are in motion. In this case we as observer can not measure the exact speed of the an object, because measured quantity of motion of other object is vary with the magnitude and direction of our motion. This can be studied with mathematical proof in this chapter. The Inertial frame and non inertial frame of reference, Special theory of relativity is covered here.

1. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 1
Theory of Relativity:
Einstein overthrew many assumptions underlying earlier physical theories, redefining in the process the fundamental
concepts of space, time, matter, energy, and gravity on the basis of new theory called Relativity. Along with quantum
mechanics, theory of relativity is also take important stage in modern physics. In particular, relativity lays foundations
for understanding cosmic processes and the geometry of the universe itself many technologies like GPS and
positioning satellites.
According to the theory relativity, motion of a body never been an absolute quantity but it is relative to an observer.
The magnitude of motion of a body moving a particular frame is depending on the magnitude and direction of
observer w.r.t to that body.
For example, consider moving train A, another train B moving on left side in same direction of train A and a stationary
observer (C) is standing right side of train A. The velocity of train A, reference to the observer in train B is not same as
that of stationary observer C. Therefore, the motion of a body is not absolute but it is relative.
Some definitions relative to the theory of relativity:
Event: An event is something that happens at a particular point in space and at a particular instant of time,
independent of the reference frame. Which we may use to described it.
Observer: An observer is a person or equipment meant to observe and take measurement about the event. The
observer is supposed to have with him scale, clock and other needful things to observe that event.
Frame of reference: A point object either at rest or in motion or any event can be described using a coordinate system.
This coordinate system is called the Frame of Reference. For example, a Cartesian coordinate system represented X, Y
and Z axes in which position of point object is shown by coordinates x, y, z respectively.
This coordinate system with X, Y, Z axes, which gives the details of point object, is called a frame of reference. For Ex.
In below fig, S is a frame of reference, a is point with coordinates (x, y, z, t) is at location A at time t, if it did not change
its location w.r.t to time then it is said to be at rest. If point changed its location to B with coordinates (x’, y’, z’, t’) then
it is said to be in motion.
In fig. 2, the two frames define the motion of the point ‘p’, and then observations of both frames need not be same
even they are explaining same point.

2. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 2
There are two frames of references 1) Inertial frame of reference and 2) Non Inertial frame of reference.
Inertial Frame of Reference:
 The frame of reference in motion relative to an observer is said to be Inertial frame of reference, only when the
law of inertia or Newton’s first law holds well in the system, i.e. there is no external force acting on the system.
 This is also called as an un-accelerated frame of reference because, the frame remains at rest or is moving with an
uniform motion along straight line. It is free from linear acceleration or rotational acceleration.
 If another frame that is moving uniformly relative to the observer’s frame is also inertial.
 Consider a train running with a uniform velocity a boy bouncing a rubber ball to the floor, even a ball remains in air
some time; it directly goes into the hand of boy, this same for the observer in train and stationary observer outside
train it happens in Inertial frames only.
 If two frames are moving relative to each other the relative velocity measured by them measured by persons both
the frames is same (velocity of one frame measured sitting in another frame)
Non-inertial Frame of reference:
 The frame of reference in motion relative to an observer is said to be Inertial frame of reference, I the law of
inertia or Newton’s first law holds good in the system, i.e. there is no external force acting on the system.
 For example, when a uniformly moving vehicle changed its speed, there would be change in the nature of
passenger, the passenger fall forward on reducing the speed or fall backward on increase in the speed.
 The force act upon passenger is called as pseudo force because there is source of existence for this force; it is
acting opposite to the direction of acceleration of frame.
 The relative motion due to non inertial frame cannot directly explained by the newton’s first law of motion.
GALILEAN TRANSFORMATION EQUATIONS:
When an event is observed two different inertial frames, then the relationship between the coordinates of two frames
are obtained in the form of equations called Galilean transformations.
Case I: For the observer O in frame ‘S’
 Consider two inertial reference frames named as S and S’ are two frames with coordinate axes XYZ and X’Y’Z’
respectively, also O and O’ are the origins of both coordinate axes.


 In inertial frame of reference, relative motion does not affect the length so we have, x’ = x – vt, along x-axis, there
is no motion along Y and Z axes, therefore y’ = y, z’ = z,
 The time at which event observed by S and S’ is t and t’ time respectively and as time is absolute quantity t = t’
 The above equations x’ = x – vt, y’ = y, z’ = z, t’ = t are called the Galilean transformation equations.

x’
x
 Consider frame S is stationary and frame S’ is moving with a uniform
velocity ‘v’ along positive X-axis. Assume the time, t=t’=0 when two
frames are coincided with each other.
 At time ‘t’ an event observed by observers in two frames, at point ‘P’, at a
distance x from frame S and at x’ from frame S’ respectively. By this time
the frame S’ may travelled through distance vt from S, as shown in figure.

3. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 3
CaseII: For observer O’ in S’
 The above transformation equations are called as inverse Galilean transformation equations.
MICHELSON MORLEY EXPERIMENT:
Michelson strongly believed that the light waves also travel at different speeds in ether relative to that in a vacuum
and any medium having a density will change the direction of light passing through it due to the phenomenon of
refraction. He had also developed an interferometer to experiment on the arriving light beams and prove his theory.
The Michelson Morley experiment was conducted to determine the presence of ether as a medium to propagate light
and to find the speed of earth. The method of theory of relativity is used here, so it also became an experiment to
verify the invariance of light.
MICHELSON MORLEY EXPERIMENT:
Michelson strongly believed that the light waves also travel at different speeds in ether relative to that in a
vacuum and any medium having a density will change the direction of light passing through it due to the
phenomenon of refraction. He had also developed an interferometer to experiment on the arriving light
beams and prove his theory.
The Michelson Morley experiment was conducted to determine the presence of ether as a medium to
propagate light and to find the speed of earth. The method of theory of relativity is used here, so it also
became an experiment to verify the invariance of light.
The fig. 1 above gives the schematic diagram of Michelson Morley Experiment. It consists of a half silvered
mirror (S) in the middle a two reflecting mirrors M1 along horizontal direction and M2 along vertical
direction from the ‘S’ at a distance of ‘d’ as shown.
Fig. 1 fig. 2
x
SM2
S
M1
M2 M2’
d
d
M2
S
M1
–
x
x’
 In this case one should think that the observer in S’ is stationary and
frame S moving backward. The dimensions of length and time are
same as Case I.
 The velocity of frame S observed by ‘S’ is –v.
 Now we can write the equation x = x’ – (–vt) , y=y’, z=z’ and t=t’. or
x = x’ + vt, y = y’, z = z’ and t = t’.

4. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 4
Consider a input beam of light from source focused on the half silvered mirror as shown in fig. 1, the single
beam splits into two beams of light one reflect in the direction of mirror M1 along horizontal direction and
other transmits through the ‘S’ and reaches mirror M2 along vertical direction. Both the beams reflect back
to ‘S’ and both travel towards the detector (R) as combined beam.
Consider the motion of the experimental system is along horizontal direction with velocity ‘v’ and velocity of
light is ‘c’
Case I: Horizontal direction: distance between S and M1 i.e., SM1 = d, during forward journey ( S M1) the
relative velocity of light is, c – v and time taken to travel is, 𝑡𝑎 =
𝑑
𝑐−𝑣
After reflection at mirror M1, beam returns to ‘S’, with relative velocity is c+v, time taken for this is 𝑡𝑏 =
𝑑
𝑐+𝑣
Time taken for complete travel S to M1 to back S is,
𝑡1 = 𝑡𝑎 + 𝑡𝑏 =
𝑑
𝑐−𝑣
+
𝑑
𝑐+𝑣
=
2𝑑
𝑐
[1 +
𝑣2
𝑐2
]
Case II: Vertical direction: here when light leave mirror ‘S’, beam cannot reach M2 in its original place (M2),
because the mirror keep moving with system, hence the beam reaches it at a distance ‘x’ from original
direction (at point M2’) as shown in fig 2.
The original distance between S and M2 i.e. SM2 = d, distance between M2 and M2’ is ‘x’, then we get, new
distance traveled by light is
(SM2’)2
= (SM2)2
+ x2
= d2
+ x2
 (cta’)2
= d2
+ (vta’)2
or (cta’)2
- (vta’)2
= d2
Or 𝒕𝒂
′𝟐
=
𝒅𝟐
𝒄𝟐−𝒗𝟐
Or 𝒕𝒂
′
=
𝒅
√𝒄𝟐−𝒗𝟐
𝒐𝒓 𝒕𝒂
′
=
𝟐𝒅
𝒄
[𝟏 +
𝒗𝟐
𝟐𝒄𝟐
]
In case of return journey after reflecting at M2 to reach ‘S’ it takes same time, i.e. 𝒕𝒃
′
=
𝒅
√𝒄𝟐−𝒗𝟐
Therefore taken to vertical journey is 𝑡2 = 𝑡𝑎
′
+ 𝑡𝑏
′
=
𝒅
√𝒄𝟐−𝒗𝟐
+
𝒅
√𝒄𝟐−𝒗𝟐
=
𝟐𝒅
√𝒄𝟐−𝒗𝟐
=
𝟐𝒅
𝒄
[𝟏 +
𝒗𝟐
𝟐𝒄𝟐
]
Hence difference in the velocity of light during horizontal and vertical journey is given by,
∆𝑡 = 𝑡1 − 𝑡2 =
2𝑑
𝑐
[1 +
𝑣2
𝑐2
] −
𝟐𝒅
𝒄
[𝟏 +
𝒗𝟐
𝟐𝒄𝟐
]
∆𝒕 =
𝟐𝒅
𝒄
[
𝒗𝟐
𝟐𝒄𝟐
] 𝒕𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆, ∆𝒕 = 𝒅 [
𝒗𝟐
𝒄𝟑
]
This is the time difference should arise between the horizontally and vertically travelled light beams while
reaching the detector.
The Michelson Morley experiment performed to show the presence of ether and hence to find correct speed
of earth moving around son. It would have succeeded if there should be time difference between the light
beams reaching detector as shown in theory. But experimental results did not shown any sign of if
difference but, t =0.0, hence it failed prove the presence of ether.

5. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 5
Secondly it failed to prove the relative velocity of light. If velocity of light follow the relativity concept then
the beams may reach detector at different time, but difference in reach is negligible. Hence Michelson
Morley experiment is considered as a failed experiment. It proved that light speed is absolute on all frames
and not involved in the relativity concept.
EINSTEIN’S SPECIAL THEORY OF RELATIVITY
Einstein proposed the special theory of relativity in 1905. This theory deals with the problems of mechanics
in which one frame moves with constant velocity relative to the other frame.
The two postulates of the Special Theory of Relativity are:
1. The laws of physics are the same in all inertial systems. No preferred inertial system exists.
2. The speed of light(c) in free space has the same value in all the inertial systems.
LORENTZ TRANSFORMATION EQUATION
Results of Galilean Transformation equations cannot be applied for the objects moving with a speed
comparative to the speed of the light.
Therefore, new transformations equations are derived by Lorentz for these objects and these are known as
Lorentz transformation equations for space and time.
Assume short comes of the Galilean transforms can be corrected by multiplying them with an independent
constant ‘K’,
Therefore, x = K (x’ + vt’) and x’ = K (x – vt) --------------------- 1
From fig. we have t2
=
x2
c2 or x2
− c2
t2
= 0, similarly, x′2
− c2
t′2
=0
From the above two equations we get, x2
− c2
t2
= x′2
− c2
t′2
----------------------- 2
Using transformation equations (1) in Equations (2) we get, the new transformation equations of S’
measured by S frame
𝑥′
=
𝑥−𝑣𝑡
√1−
𝑣2
𝑐2
, y’ = y, z’ = z and 𝑡′
=
[𝑡−
𝑣𝑥
𝑐2]
√1−
𝑣2
𝑐2
are called Lorentz transformation equations,
Similarly the Transformation equations measured by frame S’ for frane S are,
𝑥 =
𝑥′ +𝑣𝑡′
√1−
𝑣2
𝑐2
, y’ = y, z’ = z and 𝑡 =
[𝑡′+
𝑣𝑥′
𝑐2 ]
√1−
𝑣2
𝑐2
These are called inverse Lorenz transformation equations.
Merits of Lorentz transformation:
1. The Lorentz transformation equations can be in compatible with the relativistic velocity of light
v
S S’
x’
x
P
X
Y
Let there are two inertial frames of references S and S’. S is at rest
and S’ is moving w.r.t S with velocity ‘v’ along positive X-axis.
At time t=t’=0, origins O and O’ of both frames are coincided. After
time‘t’ an event happens at position P in the frame space. The
coordinates of the P will be x’ according to the observer O’ in S’ and
it will be x according to observer O in S.

6. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 6
2. The Maxwell’s electromagnetic equations are remain invariant under the Lorentz transformation
equations.
3. The invariance of velocity of light can be proved with Lorentz transformation equations.
Applications of Lorentz transformation equations:
1. Length Contraction:
Lo
The stationary observer in the frame of reference S measure this length as L, then L < Lo
Now, The length of the rod according to stationary observer in frame S, L = x2 –x1 ……………………… 1
The length of the rod according to the observer in moving frame S’, Lo = x2‘–x1’ ……………………… 2
According to Lorentz transformation,
𝑥2
′
=
𝑥2−𝑣𝑡
√1−
𝑣2
𝑐2
and 𝑥1
′
=
𝑥1−𝑣𝑡
√1−
𝑣2
𝑐2
------------------------------------------------3
From equation 1. Lo =x2’– x1’ Substitute eqn 3 in eqn 1
𝐿𝑜 =
𝑥2−𝑣𝑡
√1−𝑣2
𝑐2
−
𝑥1−𝑣𝑡
√1−
𝑣2
𝑐2
=
𝑥2−𝑥1
√1−
𝑣2
𝑐2
but L = x2 –x1
Therefore 𝐿𝑜 =
𝐿
√1−
𝑣2
𝑐2
or 𝐿𝑜 × √1 −
𝑣2
𝑐2 = 𝐿--------------------------------4
From equation 4 it is found that, v<<c, therefore,
𝑣2
𝑐2 < 1, therefore, √𝟏 −
𝒗𝟐
𝒄𝟐 < 1,
Intern this implies L’< L, this proved length appeared contracted when length measured from the moving
frame of reference.
2. Time Dilation: Consider two frames of reference, a stationary frame and a moving frame then, an
observer in stationary frame finds that, the time taken for an event in moving frame appears longer than the
same event taking place in stationary frame; this phenomenon is called time dilation.
Consider an event takes place in moving frame of reference S' and time taken to complete that event t2' – t1'.
An observer from a stationary frame of reference S, measures time taken to complete same event t2 – t1
From the Lorentz transport equation for time,
𝑡1
′
=
[𝑡1−
𝑣𝑥
𝑐2]
√1−
𝑣2
𝑐2
and 𝑡2
′
=
[𝑡2−
𝑣𝑥
𝑐2]
√1−
𝑣2
𝑐2
Now, 𝑡2
′
− 𝑡1
′
=
[𝑡2−
𝑣𝑥
𝑐2]
√1−
𝑣2
𝑐2
−
[𝑡1−
𝑣𝑥
𝑐2]
√1−
𝑣2
𝑐2
=
[𝑡2−𝑡1]
√1−
𝑣2
𝑐2
v
S S’
x’1
x1
X
Y
x’2
x2
When an object is moving relative to an observer, then length of
that object is appeared to be contracted along axis of relative
motion and inversely proportional to magnitude of relative
motion, this phenomenon is called as Length contraction.
Consider a rod of length Lo is in moving frame S’, is called as proper
length or actual length.

7. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 7
Substituting, 𝑡2
′
− 𝑡1
′
= 𝑡 and 𝑡𝑜 = 𝑡2 − 𝑡1
Therefore, 𝑡 =
𝑡𝑜
√1−
𝑣2
𝑐2
, is called equation of time dilation. When v≠0 and v < c. then t > to, i.e. time is dilated.
Relativistic Mass
The well known special theory of relativity also hints a stationary observer finds that there is an increase in the
mass of a particle moving with a relativistic velocity. This concept is relativistic mass. Comparable to length
contraction and time dilation a thing called mass increase happens when the object is in motion.
The relativistic mass formula is articulated as,
m =
m0
√1 −
v2
c2
Where, the rest mass is mo, the velocity of the moving body is v, the velocity of light is c.
Numericals:
1. A length of spaceship is measured to be half of its proper length. Find the relative velocity of the
observer.
Proper length of space ship = Lo
Improper length (length measured by observer) of spaceship is L =Lo /2
𝑳𝒐 =
𝐿
√1−
𝑣2
𝑐2
→→ 𝐿𝑜 =
𝐿𝑜 /2
√1−
𝑣2
(3𝑥108)
2
1
2
= √1 −
𝑣2
(3𝑥108)2  1
4
= 1 −
𝑣2
(3𝑥108)2
or 1 −
1
4
=
𝑣2
(3𝑥108)2 or
3
4
× (3𝑥108)2
= 𝑣2
therefore 𝑣 = 2.64 × 108
𝑚/𝑠
2. A hypothetical train moving with a speed of 0.6c passes by the platform of a small station without
being slowed down the observer on the platform notes the length of the train is just equal to the length
of platform equals to 200m. Find the real length of platform.
i) Find the real length of the train.
ii) Find the length of the platform as measured by the observer in the train.
i) V = 0.6c, L = 200m
Real length of the train is given by, 𝑳𝒐 =
𝐿
√1 −
𝑣2
𝑐2
=
200
√1−
(0.6𝑐)2
𝑐2
=
200
√1−0.62
, therefore Lo = 250m
ii) The lengths of platform measured by observe in train.
𝐿𝑜√1 −
𝑣2
𝑐2 = 𝐿 or 200√1 − (0.6)2 = 𝐿

8. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 8
Therefore L = 160m.
3. A circle of radius 5cm, lies at the rest in X-Y plane, for an observer, who is moving with uniform
velocity along the y- direction appears to be an ellipse with equation
𝒙𝟐
𝟐𝟓
+
𝒚𝟐
𝟗
= 𝟏, Find the
velocity of observer.
For moving observer the circle of radius 5cm appears to be ellipse, of equation
𝑥2
25
+
𝑦2
9
= 1
Compare above equation with
𝑥2
𝑏2 +
𝑦2
𝑎2 = 1
b = Semi major axis - Here Proper length Lo = 5m
a = Semi minor axis (decreased length along X- axis) And Improper length = 3cm
From the equation
𝐿𝑜√1 −
𝑣2
𝑐2 = 𝐿  (
𝐿
𝐿𝑜
)
2
= 1 −
𝑣2
𝑐2
1 − (
3
5
)
2
=
𝑣2
𝑐2 v = 2.4x108
m/s
4. Determine the relativistic time if an astronauts
1) One move in a spaceship with velocity 0.55c and time measured in his frame is 7years.
2) The other astronaut travels with speed 0.75c; time measured in his frame is also 7 years.
Solution 1) velocity of spaceship v = 0.55c, to= 7year, then relativistic time t is,
t =
to
√1−
v2
c2
=
7
√1−
(0.55c)2
c2
= 8.38 year
2) velocity of spaceship v = 0.75c, to= 7year, then relativistic time t is,
t =
to
√1−
v2
c2
=
7
√1−
(0.75c)2
c2
= 10.57 year
5) A particle of mass 1.67 × 10−24
kg travels with velocity 0.65c. Compute its rest mass?
Answer:
Given: Mass m = 1.67 × 10−24
kg, v = 0.65c, c = 3 × 108
m/s2.
The relativistic mass formula is articulated as,
m=m0 /√1 − v2/c2
1.67×10−24
=m0 √1 − (0.65)2c2/c2
Rest mass, mo = 1.26 × 10−24
kg (approximately). Thus, the rest mass of the particle is 1.26 × 10−24
kg

9. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 9
COMPLETE

10. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 10
The basic problem, first observed by the Michelson-Morley experiment, is that no matter how an
observer measures the local speed of light in a vacuum, the result is always c. There are two
approaches to making this fact consistent with our body of knowledge of physics. I here use a
structured list to enable point-by-point comparison of the two approaches. The two approaches are
labeled “a“ and “b“.
1. We suppose that
a. Light propagates at speed c with respect to some fixed coordinate system, hereafter
called “aether”. When an observer is moving with respect to the aether, the observer’s
velocity is subtracted from the velocity of light in accordance with Galilean relativity.
b. There is no fixed coordinate system with respect to which light propagates, or anything
else moves. Instead, space-time inherently has a geometry such that, if anything moves
with speed c as observed by any observer, any other observer, no matter how moving
in relation to that first observer, will observe that same speed, c. This is special
relativity, as proposed by Einstein.
2. In a measurement of the speed of light,
a. Whenever an observer who is moving in relation to the aether tries to measure the
speed of light, the observer and his/her equipment are affected by their own
movement in relation to the aether, in just the right way that the equipment will yield
the result c. Such effects, known collectively as the Lorentz transformation, are
i. shrinking of the observer and equipment along the direction of motion,
called FitzGerald contraction, and
ii. slowing-down of time affecting the observer and equipment, called time dilation.
b. The Lorentz transformation is just an effect of an observation from a different point of
view, which is influenced not only by change of location and rotation, but also by
velocity of the observer relative to that which is observed. Minkowski combined space
and time into four-dimensional space-time. By the neat trick of an odd definition of
distance in that four-dimensional space, Minkowski applied non-Euclidean
geometry of Riemann, simplified by the tensor calculus of Ricci-Curbastro and others,
so that the Lorentz transformation is described by the same mathematics as the varied
observations of a three-dimensional object as observed by stationary observers in
different relative locations and orientations.
3. An object has different shape according to observers with different velocities.
a. The Lorentz transformation is not entirely far-fetched. FitzGerald, after whom the
length contraction, item 2 a i is named, pointed out that the matter of which humans
and their equipment are composed is held together by electrostatic attraction between
oppositely charged electrons and protons. According to Maxwell’s equations, as
electric charges move through the aether, they create magnetic fields. FitzGerald
calculated that the magnetic fields of moving matter will make it contract along the
direction of motion just the right amount, per item 2 a i.
For example, a particularly simple argument of symmetry shows that an undisturbed

11. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 11
soap bubble should be spherical, yet a relatively moving observer will observe it
somewhat flattened along the direction of relative motion. FitzGerald’s finding is that
neither observation contradicts basic laws of physics.
I suppose that those magnetic fields would also slow down a clockwork mechanism the
right amount to conform to time dilation, although I am not aware of a direct
calculation of such an effect.
This explanation is a bit odd, because there is no particular reason why Maxwell’s
equations should participate in a mysterious conspiracy to conceal the relative velocity
of light.
b. Because of the simplicity and generality of the tensor transformations derived by
Minkowski, he was able to apply them to other phenomena besides dimension and
time changes of objects. When applied to an electric field, Minkowski’s tensor
expression of the transformation shows that a magnetic field is simply the appearance
of an electric field as observed by an observer in motion relative to the charge that is
the source of the field.
The soap bubble that is observed to be spherical by a relatively stationary observer
and flattened with added magnetic fields by a relatively moving observer are fully
consistent with each other, because they are just different views of the same thing,
related through the tensor expression of the Lorentz transformation, applied to both
the geometry and the internal forces of the bubble.
4. The shape of a solid body is set not only by electromagnetic force. In a large body of
matter, over about 400 kilometres diameter, the force of gravity becomes dominant over
the strengths of solids; any such body has a spherical shape (aside from effects of rotation),
because gravity pulls anything toward the most compact shape. Hereafter, I refer to a
“planet” for brevity, but the following is true of any such large body. One could in principle
use a planet as the basis of an instrument to measure the speed of light. We already have
done so, in effect, by combining signals from radio telescopes all over our planet to form
an effective radio telescope the size of Earth.
This means that gravitation must also conform to the Lorentz transformation. Therefore,
gravitation must also have a velocity-dependent component, like electric charge has
magnetism.
a. We therefore must suppose both electromagnetism and gravitation conspire to hide
the relative velocity of light.
This conspiracy theory is getting increasingly far-fetched. There is no particular reason
why gravitation should join the already unexplained conspiracy of electromagnetism to
obstruct attempts to measure the relative speed of light. The only direct observational
evidence for a velocity-dependent component of gravitation is that light that passes
near a gravitating body is deflected twice as much as Newtonian gravity predicts, as

12. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 12
was famously observed during solar eclipse of 1919. That observation gives little
guidance for constructing a velocity-dependent component of gravitation.
b. The source term for Newtonian gravitation is mass. When Einstein’s equivalence of
mass and energy is combined with Minkowski’s tensor formulation of the Lorentz
transformation, mass becomes a tensor quantity that combines mass, energy,
momentum, and stress, which credibly introduces velocity dependence to gravitation.
The tensor expression of space-time also enables description of gravitational effects on
space time in terms of Ricci-Curbastro’s metric tensor and curvature tensor. These
concepts, along with much work, enabled Einstein to discover the theory of general
relativity, which indeed shows the appropriate changes in shape of a gravitationally
formed body, and which yields many other distinctive results that have been confirmed
experimentally.
5. One can in principle suppose experiments in which the nuclear force and the weak force are
used as standards for measurement of the speed of light. In particular, the shapes of atomic
nuclei are determined by a combination of electrostatic and “strong” force.
a. To sustain the conspiracy to hide the “true” relative speed of light, the strong force
must have a velocity-dependent component that matches that of electrostatic force. At
the time that the two approaches were under discussion, these forces were unknown,
so no particular reason has been adduced for them to join the conspiracy.
b. Quantum field theory is our best explanation of the strong and weak forces, and is
made relativistically correct simply by the fact that it is formulated using Minkowski’s
application of tensor analysis to the four-dimensional space time in which the field
theory is set. (Quantum field theory and general relativity are at present mutually
inconsistent in certain respects, but that does not affect the consistency of nuclear
forces with special relativity.)
Thus, approach a is seen to be a rather improbable concatenation of coincidences, while approach b
is a self-consistent body of rigorous knowledge by considering the invariance of c to be an inherent
property of space-time. The question of why c is invariant thus becomes part of the question of why
space-time has the properties it has, along with the question of e.g. why space-time has three
(observable) spatial dimensions and one time dimension.
The reasons for these and other fundamental properties of the Universe could in principle be
explained by derivation from some more fundamental principle. An example of that sort of
explanation is Noether’s Theorem, which derives conservation of energy and momentum by the
assertion that the laws of physics are the same everywhere and everywhen (and similarly for other
conservation laws).
Even if an equivalent theorem is waiting to be discovered, that would similarly derive the nature of
space-time, including the invariance of c, from another principle, ultimately there must exist one or
more principles that are unexplained. Thus, there is no possibility, even in principle, of a much more
satisfactory explanation of the invariance of c than the assertion that it is a fundamental law of

13. “Theory of Relativity”. Engineering Physics Study Materials by Praveen N. Vaidya Page 13
physics, especially when that one fundamental law sews up in a clear explanation so many otherwise
untidy characteristics of the Universe.