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Source: Postulates of special relativity

Albert Einstein derived the theory of special relativity in 1905, from principle now called the postulates of special relativity. Einstein's formulation is said to only require two postulates, though his derivation implies a few more assumptions.
The idea that special relativity depended only on two postulates, both of which seemed to follow from the theory and experiment of the day, was one of the most compelling arguments for the correctness of the theory (Einstein 1912: "This theory is correct to the extent to which the two principles upon which it is based are correct. Since these seem to be correct to a great extent, ...")

Postulates of special relativity

relativity

special

relativity

postulates

Alternative derivations of special relativity

Insufficiency of the two standard postulates

The 1905 system as "null theory"

Alternative relativistic models

Mathematical formulation of the postulates

special

relativity

with the following property. If A, B are two events which have coordinates

(

x

1

,

x

2

,

x

3

,
t
)

{\displaystyle (x_{1},x_{2},x_{3},t)}

and

(

y

1

,

y

2

,

y

3

,
s
)

{\displaystyle (y_{1},y_{2},y_{3},s)}

in one inertial frame

F

{\displaystyle F}

, and have coordinates

(

x

1

′

,

x

2

′

,

x

3

′

,

t
′

)

{\displaystyle (x'_{1},x'_{2},x'_{3},t')}

and

(

y

1

′

,

y

2

′

,

y

3

′

,

s
′

)

{\displaystyle (y'_{1},y'_{2},y'_{3},s')}

in another inertial frame

F
′

{\displaystyle F'}

, then

(

x

1

−

y

1

)

2

+
(

x

2

−

y

2

)

2

+
(

x

3

−

y

3

)

2

=
c
(
s
−
t
)

{\displaystyle {\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}=c(s-t)\quad }

if and only if

(

x

1

′

−

y

1

′

)

2

+
(

x

2

′

−

y

2

′

)

2

+
(

x

3

′

−

y

3

′

)

2

=
c
(

s
′

−

t
′

)

{\displaystyle \quad {\sqrt {(x'_{1}-y'_{1})^{2}+(x'_{2}-y'_{2})^{2}+(x'_{3}-y'_{3})^{2}}}=c(s'-t')}

.
Informally, the Second Postulate asserts that objects travelling at speed c in one reference frame will necessarily travel at speed c in all reference frames. This postulate is a subset of the postulates that underlie Maxwell's equations in the interpretation given to them in the context of special relativity. However, Maxwell's equations rely on several other postulates, some of which are now known to be false (e.g., Maxwell's equations cannot account for the quantum attributes of electromagnetic radiation).
The second postulate can be used to imply a stronger version of itself, namely that the spacetime interval is invariant under changes of inertial reference frame. In the above notation, this means that

c

2

(
s
−
t

)

2

−
(

x

1

−

y

1

)

2

−
(

x

2

−

y

2

)

2

−
(

x

3

−

y

3

)

2

{\displaystyle c^{2}(s-t)^{2}-(x_{1}-y_{1})^{2}-(x_{2}-y_{2})^{2}-(x_{3}-y_{3})^{2}}

=

c

2

(

s
′

−

t
′

)

2

−
(

x

1

′

−

y

1

′

)

2

−
(

x

2

′

−

y

2

′

)

2

−
(

x

3

′

−

y

3

′

)

2

{\displaystyle =c^{2}(s'-t')^{2}-(x'_{1}-y'_{1})^{2}-(x'_{2}-y'_{2})^{2}-(x'_{3}-y'_{3})^{2}}

for any two events A, B. This can in turn be used to deduce the transformation laws between reference frames; see Lorentz transformation.
The postulates of special relativity can be expressed very succinctly using the mathematical language of pseudo-Riemannian manifolds. The second postulate is then an assertion that the four-dimensional spacetime M is a pseudo-Riemannian manifold equipped with a metric g of signature (1,3), which is given by the Minkowski metric when measured in each inertial reference frame. This metric is viewed as one of the physical quantities of the theory; thus it transforms in a certain manner when the frame of reference is changed, and it can be legitimately used in describing the laws of physics. The first postulate is an assertion that the laws of physics are invariant when represented in any frame of reference for which g is given by the Minkowski metric. One advantage of this formulation is that it is now easy to compare special relativity with general relativity, in which the same two postulates hold but the assumption that the metric is required to be Minkowski is dropped.
The theory of Galilean relativity is the limiting case of special relativity in the limit

c
→
∞

{\displaystyle c\to \infty }

(which is sometimes referred to as the non-relativistic limit). In this theory, the first postulate remains unchanged, but the second postulate is modified to:

If A, B are two events which have coordinates

(

x

1

,

x

2

,

x

3

,
t
)

{\displaystyle (x_{1},x_{2},x_{3},t)}

and

(

y

1

,

y

2

,

y

3

,
s
)

{\displaystyle (y_{1},y_{2},y_{3},s)}

in one inertial frame

F

{\displaystyle F}

, and have coordinates

(

x

1

′

,

x

2

′

,

x

3

′

,

t
′

)

{\displaystyle (x'_{1},x'_{2},x'_{3},t')}

and

(

y

1

′

,

y

2

′

,

y

3

′

,

s
′

)

{\displaystyle (y'_{1},y'_{2},y'_{3},s')}

in another inertial frame

F
′

{\displaystyle F'}

, then

s
−
t
=

s
′

−

t
′

{\displaystyle s-t=s'-t'}

. Furthermore, if

s
−
t
=

s
′

−

t
′

=
0

{\displaystyle s-t=s'-t'=0}

, then

(

x

1

−

y

1

)

2

+
(

x

2

−

y

2

)

2

+
(

x

3

−

y

3

)

2

{\displaystyle \quad {\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}}

=

(

x

1

′

−

y

1

′

)

2

+
(

x

2

′

−

y

2

′

)

2

+
(

x

3

′

−

y

3

′

)

2

{\displaystyle ={\sqrt {(x'_{1}-y'_{1})^{2}+(x'_{2}-y'_{2})^{2}+(x'_{3}-y'_{3})^{2}}}}

.
The physical theory given by classical mechanics, and Newtonian gravity is consistent with Galilean relativity, but not special relativity. Conversely, Maxwell's equations are not consistent with Galilean relativity unless one postulates the existence of a physical aether. In a number of cases, the laws of physics in special relativity (such as the equation

E
=
m

c

2

{\displaystyle E=mc^{2}}

) can be deduced by combining the postulates of special relativity with the hypothesis that the laws of special relativity approach the laws of classical mechanics in the non-relativistic limit.