Spacetime

A **spacetime**, also known as a **spacetime continuum** of a universe is the space of events in said universe. The “distance” between two events (points) analogous to the Pythagorean theorem (

(
d
s

)
2

=
(
d
x

)
2

(
d
y

)
2

(
d
z

)
2

{\displaystyle (ds)^2 = (dx)^2 + (dy)^2 + (dz)^2}

, for 3D Euclidean realm) from a Euclidean space in a spacetime is called the **spacetime interval** and in special relativity, the spacetime interval follows the Minkowski metric (

(
d
s

)
2

=
(
d
x

)
2

(
d
y

)
2

(
d
z

)
2

−
(
d
t

)
2

{\displaystyle (ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2}

 or

(
d
s

)
2

=
(
d
t

)
2

−
(
d
x

)
2

−
(
d
y

)
2

−
(
d
z

)
2

{\displaystyle (ds)^2 = (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2}

, for 4D Minkowski flune, 3 spacelike (spatial) dimensions and 1 timelike (temporal) dimension). This is to keep the speed of massless particles a constant. According to general relativity, the spacetime interval actually changes when a mass-energy is present, how much so is found from Einstein’s field equations. Intuitively, this spacetime curves or “warps” when matter is present. This curvature is the cause of gravity, which is not a force according to general relativity, but just objects travelling in a geodesic along spacetime.

# Transcendental Aesthetic

Immanuel Kant viewed Space and time as the two sensible intuitions, space being external and time being internal

## Space

Kant viewed Space as the outer sense which is a priori.

He deduced that Space is a priori out of 4 reasons:

1. To be able to observe external sensations space must already exist
2. One couldn't imagine that space is not
3. Space is the whole and individual spaces are merely parts of the whole space
4. Space is imagined to be infinite

## Time

Kant viewed time not as something external, but more as something which is internal in us.

He deduced that time is a priori out of 4 reasons:

1. One couldn't imagine succession without time
2. One couldn't imagine that time is not
3. Time is the whole and individual time periods are merely a part of the whole time
4. The original idea of time is unrestricted

# Relativity

## Special relativity

A spacetime diagram is often used in special relativity to show relativistic effects of moving observers such as length contraction and time dilation from Lorentz transformations. In special relativity, spacetime is a perfectly flat Minkowski spacetime. The correct way to view things from a moving inertial perspective is through a hyperbolic rotation, a Lorentz transformation. The point at

(
t
,
x
,
y
,
z
)

{\displaystyle (t, x, y, z)}

 moves to the point

(
γ
(
t
−

v

x

c
2

)
,
γ
(
x
−

v

t
)
,
y
,
z
)

{\displaystyle (\gamma (t - \frac {{\bf{v}} x}{c^2}), \gamma (x - {\bf{v}} t),y,z)}

 where

γ

{\displaystyle \gamma}

 is the Lorentz factor

1

1
−

v

2

c
2

{\displaystyle \frac {1}{\sqrt {1 - \frac {{\bf {v}}^2}{c^2}}}}

 and

c

{\displaystyle c}

 is the speed of light when a Lorentz transform is applied to view the perspective of an observer traveling at velocity

v

{\displaystyle \bf {v}}

. This is in stark constrast to the Galilean transformation used in classical mechanics where space and time are not part of a continuum.

A Minkowski spacetime adjoined with all Lorentz transformations for all velocities between

−
c

{\displaystyle -c}

 and

c

{\displaystyle c}

 form a group called the Lorentz group. A Minkowski plane with its Lorentz group is isomorphic to the split-complex line adjoined by multiplication of elements on the left branch of the unit hyperbola, finding out where an event’s coordinates (x,t) move to after performing a Lorentz transformation for velocity

v

{\displaystyle \bf {v}}

 is essentially finding the product of the split-complex number

x
t
j

{\displaystyle x+tj}

 and

1

1
−

v

2

−

v

1
−

v

2

j

{\displaystyle \frac{1}{\sqrt{1-{\bf {v}}^2}}-\frac{\bf {v}}{1-{\bf {v}}^2}j}

. Adjoining translations to the Lorentz group creates the Poincare group.

A caveat of special relativity Is that the Minkowski spacetime is static like the Euclidean space used in classical mechanics. General relativity changes that.

## General relativity

General relativity’s spacetime is in the form of a Lorentzian manifold, a pseudo-Riemannian manifold with one dimension’s signature different to the rest (the timelike dimension). The presence of mass-energy curves this spacetime and the relationship is found through Einstein’s field equations.