Relativity

*In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. E. T. Whittaker has pointed out that the principle of relativity is tantamount to the arbitrariness of what hyperbola radius is selected for time in the Minkowski diagram.*
From Wikipedia:Theory of relativity

Special relativity is a theory of the structure of spacetime. It was introduced in Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" (for the contributions of many other physicists see History of special relativity). Special relativity is based on two postulates which are contradictory in classical mechanics:

1. The laws of physics are the same for all observers in any inertial frame of reference relative to one another (principle of relativity).
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source.

The resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment. Moreover, the theory has many surprising and counterintuitive consequences. Some of these are:

* Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion.
* Time dilation: Moving clocks are measured to tick more slowly than an observer's "stationary" clock.
* Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer.
* Maximum speed is finite: No physical object, message or field line can travel faster than the speed of light in a vacuum.
	+ The effect of Gravity can only travel through space at the speed of light, not faster or instantaneously.
* Mass–energy equivalence: *E* = *mc*2, energy and mass are equivalent and transmutable.
* Relativistic mass, idea used by some researchers.

The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism).

## Relativity of simultaneity

Most confusion about relativity centers around a poor understanding of relativity of simultaneity.

The atoms of a moving ship really do become length contracted. This is not an illusion. Relativity does not explain this contract but it does require it. But why then does a length contracted person on the moving ship perceive that it is the rest of the universe that is length contracted when they should perceive the rest of the universe to be longer not shorter? The answer is that they perceive the universe to be shorter due to an illusion caused by relativity of simultaneity.

Since the length of an object is the distance from head to tail at one simultaneous moment, it follows that if two observers disagree about what events are simultaneous then they will also disagree about the length of objects.

If a line of clocks appear synchronized to a stationary observer and appear to be out of sync to that same observer after accelerating to a certain velocity then it follows that during the acceleration the clocks ran at different speeds. Some may even run backwards. This line of reasoning leads to general relativity. See also Relativistic rocket.

## Example

From Physics Forums

We have a particle moving with a velocity

v
→

=
v

e
→

x

{\displaystyle {\vec {v}}=v{\vec {e}}\_{x}}

 wrt. to an inertial frame

Σ

{\displaystyle \Sigma}

 and we want to know its velocity in an inertial frame

Σ
′

{\displaystyle \Sigma '}

 which moves with a velocity

w
→

=
w

e
→

y

{\displaystyle {\vec {w}}=w{\vec {e}}\_{y}}

 where v and w are equal.

It's easier to work first with four-vectors. The particle's four-velocity components wrt. to

Σ

{\displaystyle \Sigma}

 are:

u

μ

=

γ

v

(
1
,

β

v

,
0
,
0
)

=
(

γ

v

,

γ

v

β

v

,
0
,
0
)

{\displaystyle {\begin{aligned}u^{\mu }&=\gamma \_{v}(1,\beta \_{v},0,0)\\&=(\gamma \_{v},\gamma \_{v}\beta \_{v},0,0)\end{aligned}}}

Its clock is ticking at

γ
v

{\displaystyle \gamma\_v}

Its velocity is

β

v

{\displaystyle \beta \_{v}}

Its proper velocity is

γ

v

β

v

{\displaystyle \gamma \_{v}\beta \_{v}}

The components wrt.

Σ
′

{\displaystyle \Sigma '}

 are given by a Lorentz boost in

y

{\displaystyle y}

-direction with velocity

w
→

{\displaystyle \vec{w}}

, i.e., for the four-vector components you have

u

′
μ

=
[

γ

w

(

u

0

−

β

w

u

2

)
,

u

1

,

γ

w

(
−

β

w

u

0

u

2

)
,

u

3

]

=

γ

v

(

γ

w

,

β

v

,
−

β

w

γ

w

,
0
)
.

=
(

γ

v

γ

w

,

γ

v

β

v

,
−

γ

v

β

w

γ

w

,
0
)
.

{\displaystyle {\begin{aligned}u^{\prime \mu }&=[\gamma \_{w}(u^{0}-\beta \_{w}u^{2}),u^{1},\gamma \_{w}(-\beta \_{w}u^{0}+u^{2}),u^{3}]\\&=\gamma \_{v}(\gamma \_{w},\beta \_{v},-\beta \_{w}\gamma \_{w},0).\\&=(\gamma \_{v}\gamma \_{w},\gamma \_{v}\beta \_{v},-\gamma \_{v}\beta \_{w}\gamma \_{w},0).\end{aligned}}}

The first thing to note is that the resulting vector does not point halfway between

e
→

x

{\displaystyle \vec{e}\_x}

 and

e
→

y

{\displaystyle \vec{e}\_y}

. It points almost directly toward

e
→

y

{\displaystyle \vec{e}\_y}

The three-velocity thus is

v
→

′

=
c

u
→

′

/

u

′
0

=

c

γ

v

γ

w

(

γ

v

β

v

,
−

γ

v

β

w

γ

w

,
0
)

=
(
v

/

γ

w

,
−
w
,
0
)
.

{\displaystyle {\begin{aligned}{\vec {v}}'&=c{\vec {u}}'/u^{\prime 0}\\&={\frac {c}{\gamma \_{v}\gamma \_{w}}}(\gamma \_{v}\beta \_{v},-\gamma \_{v}\beta \_{w}\gamma \_{w},0)\\&=(v/\gamma \_{w},-w,0).\end{aligned}}}

Its clock is ticking at

γ

v

γ

w

{\displaystyle \gamma \_{v}\gamma \_{w}}

The

e

x

{\displaystyle e\_{x}}

 component of its velocity is

v

/

γ

w

{\displaystyle v/\gamma \_{w}}

The

e

x

{\displaystyle e\_{x}}

 component of its proper velocity is

γ

v

β

v

{\displaystyle \gamma \_{v}\beta \_{v}}

The second thing to note is that the

e

x

{\displaystyle e\_{x}}

 component of its velocity slowed down by

γ

w

{\displaystyle \gamma \_{w}}

 even though the observer accelerated perpendicular to that component. Exactly the amount its internal clock slowed down! (As measured by the observer).

The third thing to note is that even though this component of its velocity decreased neither its proper velocity nor its momentum in that same direction changed at all.

## Energy

Consider a unit force acting on a particle of unit mass. The particle is at rest at time=0.
To the left is a graph of of such a particle. The horizontal axis is nonrelativistic force \*time. Multiplying the horizontal axis by Mass gives the relativistic momentum.

The vertical axis is nonrelativistic velocity. Velocity \* dt = the distance traveled under the given force during the time interval dt. So the area under the curve is force \* distance = energy. This is true both classically (green) and relativistically (red).

Classically the kinetic energy is given by

# E

m

v

2

2

{\displaystyle E={\frac {mv^{2}}{2}}}

Relativistically the kinetic energy is given by

# E
m
(
γ
−
1
)

{\displaystyle E=m(\gamma -1)}

## Gravitational time dilation

From Gravitational time dilation

The gravitational time dilation at any point in a gravity well is equal to the time dilation that an object falling to that point would experience due its velocity (which never reaches "c") alone.

Time dilation in a gravitational field is equal to time dilation in far space, due to a speed that is needed to escape that gravitational field. Here is the proof:

1. Time dilation inside a gravitational field g per this article is

# τ
t

1
−

2
G
M

r

c

2

{\displaystyle \tau =t{\sqrt {1-{\frac {2GM}{rc^{2}}}}}}

2. **Nonrelativistic** Escape velocity from g is

# v

2
G
M

/

r

{\displaystyle v={\sqrt {2GM/r}}}

3. Time dilation formula per special relativity is

# τ
t

1
−

v

2

/

c

2

{\displaystyle \tau =t{\sqrt {1-v^{2}/c^{2}}}}

4. Substituting escape velocity for v in the above

# τ
t

1
−

2
G
M

r

c

2

{\displaystyle \tau =t{\sqrt {1-{\frac {2GM}{rc^{2}}}}}}

This should be true for any gravitational fields considering simple scenarios like non-rotation etc.

Setting escape velocity = c in 2. gives the Schwarzschild radius

r

s

=
2
G
M

/

c

2

{\displaystyle r\_{s}=2GM/c^{2}}

Substituting Schwarzschild radius in 4. we get

# τ
t

1
−

r

s

/

r

{\displaystyle \tau =t{\sqrt {1-r\_{s}/r}}}

A) Time stops at surface of a black hole since

# r

r
s

{\displaystyle r=r\_s }

 and gives

# τ
t

1
−

r

s

/

r

=
0

{\displaystyle \tau =t{\sqrt {1-r\_{s}/r}}=0}

B) Escape velocity from surface of a black hole is c, since replacing Schwarzschild radius in 2. gives

# v

2
G
M

c

2

/

2
G
M

=

c

2

=
c

{\displaystyle v={\sqrt {2GMc^{2}/2GM}}={\sqrt {c^{2}}}=c}

.
C) Time stops at speed c, since replacing v in 3. gives

# τ
t

1
−

c

2

/

c

2

=
0

{\displaystyle \tau =t{\sqrt {1-c^{2}/c^{2}}}=0}

Where:
\*

τ

{\displaystyle \tau}

 is the proper time between events A and B for a slow-ticking observer within the gravitational field,
\*

t

{\displaystyle t}

 is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object,
\*

G

{\displaystyle G}

 is the Gravitational constant,
\*

M

{\displaystyle M}

 is the mass of the object creating the gravitational field,
\*

r

{\displaystyle r}

 is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object,
\*

r
s

{\displaystyle r\_s }

 is the Schwarzschild radius,
\*

c

{\displaystyle c}

 is the speed of light,
\*

v

{\displaystyle v}

 is the velocity,
\*

g

{\displaystyle g}

 is gravitational acceleration/field =

G
M

/

r
2

{\displaystyle GM/r^2}

,
## Loophole

There might be a loophole to the law that you can't travel faster than light:
If the distance between the front of rocket and the back can be made zero then it's conceivable that it could travel much faster than light.