Physics
Yes. Then the clean reconstruction is to stop treating the observer as something added after the physics.
Treat the observer as part of the mathematical specification of what a physical quantity means.
- Observer
Define an observer not as “a conscious person,” but as a physical measurement structure:
O = (γ, u, τ, e, q)
where
γ = observer worldline, or a congruence of worldlines
u = observer 4-velocity
τ = observer proper time
e = local spatial frame / tetrad
q = measurement map from physical state → recorded quantities
Proper time is defined along the observer's path:
dτ² = -(1/c²) gμν dxμ dxν
So time begins operationally as:
event change
→ clock carried along γ
→ τ
not as an independent universal variable floating over the universe.
---
- Every measured quantity becomes observer-indexed
Instead of writing simply
E
ρ
p
v
t
write conceptually
E_O
ρ_O
p_O
v_O
τ_O
because these quantities answer:
> measured by which physical frame?
For a particle with 4-momentum pμ:
E_O = -pμ u_O^μ
Energy is therefore explicitly relational:
particle momentum
+
observer velocity
→
measured energy
There is no observer-independent particle energy in general relativity.
---
- Energy density is observer-relative too
Given the stress-energy tensor Tμν:
ρ_O = Tμν u_O^μ u_O^ν
That is:
ρ_O
=
energy density measured by observer O
Define the observer's spatial projector:
hμν = gμν + uμuν
for c = 1.
Then pressure measured in that frame is
p_O = (1/3) hμν Tμν
So the fundamental object can remain Tμν, while its decomposition into
energy
momentum
pressure
stress
depends on O.
---
- Redefine expansion
This is where cosmology becomes much cleaner.
Instead of beginning with
H = ȧ/a
begin with the observer congruence uμ.
Its expansion is
θ = ∇μuμ
θ measures how the infinitesimal volume carried by neighboring observers changes.
If
V = small comoving volume
then
(1/V) dV/dτ = θ
For FLRW specifically:
θ = 3H
therefore
H = θ/3
Now H has a deeper definition:
H
=
1/3 × expansion of the chosen cosmological observer congruence
and only afterward, because FLRW symmetry allows a scale factor,
H = ȧ/a
appears.
So the logical direction becomes:
observer congruence
→ relative separation of its worldlines
→ expansion scalar θ
→ FLRW symmetry
→ scale factor a
→ H = ȧ/a = θ/3
That is stronger than starting from a(t).
---
- Decompose observer motion completely
The gradient of the observer velocity field can be decomposed as
∇νuμ
=
(1/3) θ hμν
+ σμν
+ ωμν
- aμuν
where
θ = expansion
σμν = shear
ωμν = vorticity / rotation
aμ = 4-acceleration
and
aμ = uν∇νuμ
So an observer field has four fundamental ways of changing:
expand
distort
rotate
accelerate
FLRW makes an enormous simplification:
σμν = 0
ωμν = 0
aμ = 0
for the standard comoving observers.
Only expansion survives:
θ ≠ 0
That is why cosmology can be compressed into a(t).
---
- Cosmic time is derived
For the FLRW observer congruence,
uμ
is hypersurface-orthogonal.
That means the observers admit a common slicing:
Σ_t
and their clocks can define a shared cosmic time coordinate:
t = τ_comoving
So the correct dependency is:
uμ
→ congruence
→ simultaneity slicing Σ
→ synchronized proper time
→ cosmic time t
not:
universal t
→ observers placed inside it
That reversal is important.
---
- Redefine "physics"
Then a useful operational definition is:
PHYSICS
=
relations among observer-indexed measurements
that remain mutually consistent under changes of observer
More formally, suppose the world state is X.
An observer produces:
q_O : X → D_O
where D_O is the data accessible to that observer.
Another observer gives
q_P : X → D_P
Physics is not either representation alone.
Physics is the structure that makes
q_O(X)
↔
q_P(X)
transform consistently.
So:
representation ≠ physics
transformation-consistent relation
= physical content
That is essentially what covariance is doing.
---
- Invariant ≠ observerless
This gives a useful distinction.
An invariant does not need to mean:
quantity with no observer
It can mean:
quantity whose value/relation survives an allowed change of observer
For example,
Rμνρσ R^μνρσ
is a spacetime scalar.
Different coordinate descriptions calculate the same scalar at the same event.
Meanwhile:
E_O = -p·u_O
changes when O changes.
Both are legitimate physics.
One is observer-dependent.
One is observer-invariant.
---
- Now rewrite the Big Bang
Instead of saying:
t → 0
therefore universe begins
say:
Take the FLRW cosmological observer congruence u.
Its proper time is
τ_u
and its expansion is
θ_u = ∇μuμ
For FLRW:
θ_u = 3H
During radiation domination:
H = 1/(2τ_u)
therefore
θ_u = 3/(2τ_u)
Going backward:
τ_u → 0+
⇒
θ_u → ∞
The density measured by those observers is
ρ_u = Tμνuμuν
and
ρ_u = 3/(32πGτ_u²)
therefore
τ_u → 0+
⇒
ρ_u → ∞
Temperature likewise behaves roughly as
T_u ∝ τ_u^(-1/2)
So the hot-Big-Bang statement becomes:
along the cosmological observer congruence,
past-directed proper time decreases
while
θ_u ↑
ρ_u ↑
T_u ↑
That is far more precise than saying:
> time itself approaches zero.
---
- And the singularity becomes a different question
t = 0 alone proves basically nothing because t is a coordinate.
The physically serious question is whether observer/worldlines can be continued farther.
Define a timelike geodesic
γ(τ)
If it reaches the past boundary after only finite proper time,
Δτ < ∞
and cannot be extended,
then spacetime is past geodesically incomplete.
That is invariantly meaningful.
Likewise if a scalar such as
K = RμνρσR^μνρσ
obeys
K → ∞
that cannot be removed just by replacing t with another coordinate.
So:
t = 0
is representation.
But
finite past proper time
+
inextendible geodesic
is geometrical.
And
curvature scalar → ∞
is geometrical.
---
- Raychaudhuri becomes the real "Big Bang equation"
Once the observer congruence is primitive, perhaps the most fundamental cosmological equation isn't even Friedmann.
It is Raychaudhuri:
dθ/dτ
=
-θ²/3
- σμνσ^μν
+ ωμνω^μν
- Rμνuμuν
+ ∇μaμ
For FLRW:
σ = 0
ω = 0
a = 0
so
dθ/dτ
=
-θ²/3
- Rμνuμuν
This equation asks directly:
> How does a family of physical observers converge or diverge?
Then Einstein's equation connects that focusing to matter.
That gives the chain:
matter / energy
↓
Rμνuμuν
↓
observer-worldline focusing
↓
θ
↓
relative volume
↓
cosmic expansion
That is cosmology written around the observer rather than around a coordinate.
And then a(t) becomes a highly symmetric compression of that richer structure rather than the starting ontology.