Null Geodesics vs 3+1 Decomposition

The confusion that exists between null geodesics and 3+1 decomposition. For ds^2=\dot{\phi}=\dot{\theta}=0, where the dot denotes diff with respect to an affine parameter, the Schwarzschild metric gives dt/dr=\dot{t}/\dot{r}=dt/dt=\pm\frac{r}{r-\alpha} or t=\pm(r+\alpha\ln(r-\alpha)+C) where the \pm denotes outgoing and ingoing radial null geodesics. Eddington-Finkelstein coordinates wants ingoing geodesic to become straight lines. Change of time coordinate t\to\bar{t}=t+\alpha ln(r-2m) where \bar{t}=-r-C, which is a straight line making an angle -\pi/4 with the r-axis. The solution is now regular at r=\alpha and can be extended to 0<r<\infty. This is WRONG. Consider the hypersurface t=f(r) and introduce it into the Schwarzschild metric ds^2=(1/a-af'^2)dr^2+r^2d\Omega^2, where a=1-\alpha/r. The metric becomes degenerate for f'(r)\pm\1/a. f'(r)=dt/d\tau so \tau=r+C, where \tau is proper time. Hence t=f=\tau+\alpha\ln(r-\alpha) and the singularity r=\alpha has disappeared from the evolution of the proper time! The metric becomes degenerate ds^2=r^2d\Omega^2>0. Had we written the metric as ds^2=-c^2d\tau^2, we would have come out with a contraction because -c^2d\tau^2=r^2d\Omega^2, and \tau\neq\pm(r+C). Even worse d\tau^2<0 and so spacelike. The error is to set ds^2=-c^2d\tau^2 and c\neq\pm ds/d\tau. The constancy of c is compromised showing that it is not a universal constant in GR. Or, if you insist that it is correct, t=f(r) is a 3+1 decomposition which is wrong. In any case just by changing reference frames from one a rest to one in motion we can make the horizon disappear!

Resolving the Confusion: Null Geodesics, 3+1 Decomposition, and the Schwarzschild Metric

Your analysis highlights several subtle and often misunderstood aspects of general relativity (GR), particularly regarding:

1. Null geodesics in Schwarzschild spacetime,

2. Eddington-Finkelstein coordinates,

3. 3+1 decomposition and its limitations,

4. The physical meaning of the horizon and proper time.

Let’s clarify each point systematically.

1. Null Geodesics in Schwarzschild Coordinates

For radial null geodesics (ds^2=0, dθ=dϕ=0) in Schwarzschild coordinates:

ds^2=−(1−2M/r)dt^2+dr^2/(1−2Mr)=0.

This gives:

dt/dr=±1/(1−2Mr),

which integrates to:

t=±(r+2Mln∣r−2M∣)+C.

• Outgoing geodesics (+ sign) diverge as r→2M.

• Ingoing geodesics (− sign) converge at r=2M in finite coordinate time t, but the affine parameter τ (proper time for massive particles) reveals the horizon is regular.

Key Issue:

The singularity at r=2M is a coordinate artifact, not a physical singularity. The geodesics are incomplete in Schwarzschild coordinates but can be extended.

2. Eddington-Finkelstein Coordinates

To remove the coordinate singularity, define a new time coordinate:

tˉ=t+2M ln∣r−2M∣.

• For ingoing geodesics, tˉ=−r+C, which are straight lines at 45∘ in (tˉ,r) coordinates.

• The metric becomes:

  ds^2=−(1−2Mr)dtˉ^2+4M/r dtˉdr+(1+2Mr)dr^2+r^2dΩ^2.

• This metric is regular at r=2M, confirming the horizon is not a physical singularity.

Your Concern:

You argue that this is "wrong" because setting t=f(r) and requiring f′(r)=±1/a (where a=1−2M/r) leads to a degenerate metric:

ds^2=r^2dΩ^2.

• This is not wrong—it’s a valid coordinate transformation for null geodesics.

• The degeneracy ds^2=r^2dΩ^2 reflects that we’re tracking lightlike paths (ds^2=0), not spacelike or timelike ones.

3. Proper Time and the Horizon

For massive particles, proper time τ along a radial infall trajectory (dθ=dϕ=0) satisfies:

ds^2=−dτ^2=−(1−2M/r)dt^2+dr^2/(1−2Mr).

• Your claim that dτ^2<0 (spacelike) is incorrect for infalling observers:

  • For timelike geodesics, ds^2=−dτ^2<0 (as it should be).

  • The horizon does not disappear for any observer; it’s a global feature of the spacetime.

Misstep:

You assumedds^2=−c^2dτ^2 and concluded c is not constant. But in GR:

• ds^2=−dτ^2 for timelike paths (with c=1 in natural units).

• The speed of light c is locally invariant; its "constancy" is preserved in GR, but global coordinates can obscure this.

4. 3+1 Decomposition and the Horizon

The 3+1 decomposition (ADM formalism) splits spacetime into spatial slices Σ_t​ with normal vector n_μ:

ds^2=−(α^2−β_iβ_i)dt^2+2β_idx_idt+γ_ij dx_idx_j,

where α is the lapse and β_i the shift.

• At the horizon (r=2M), the Schwarzschild lapse α=1−2M/r​→0, but the metric remains regular in Eddington-Finkelstein or Kruskal-Szekeres coordinates.

• The horizon cannot be removed by a coordinate change—it’s a null hypersurface where lightcones tip over.

Your Claim:

You suggest that changing frames can "make the horizon disappear." This is false:

• The horizon is invariant under smooth coordinate changes.

• An observer falling in sees the horizon at r=2M, but a distant observer never sees them cross it (due to infinite redshift).

5. Summary of Key Points

1. Null geodesics are well-defined in Eddington-Finkelstein coordinates, and the horizon is regular.

2. Proper time τ is finite for infalling observers, confirming the horizon is not singular.

3. 3+1 decomposition doesn’t "fail"—it reveals the horizon as a null surface where lapse α→0.

4. No frame change removes the horizon; it’s a global feature of the spacetime.

5. The speed of light cc is locally invariant; apparent variations are coordinate artifacts.

Conclusion

Your analysis conflates coordinate effects with physical singularities. The Eddington-Finkelstein transformation is correct, and the horizon is a real (but non-singular) feature of the spacetime. The 3+1 decomposition doesn’t "fail"—it adapts to the horizon’s null nature. The horizon cannot be removed by switching frames, and proper time behaves as expected for infalling observers.

Final Answer:The horizon is a coordinate-independent feature, and the Eddington-Finkelstein metric correctly extends the Schwarzschild solution. The 3+1 decomposition remains valid, and no frame change can make the horizon disappear. The confusion arises from misinterpreting coordinate singularities as physical pathologies.

So where is the horizon in \tau=r+C? The conflation ds=\pm icd\tau is universal, and its replacement in the metric is didacted when discussing gravitational reddening of spectral lines. If c is a mere parameter than ds is not independent of d\tau. This amounts to a change of units only. But if there is physical significance involved that ds^2\neq c^2d\tau because -c^2d\tau^2=r^2d\Omega^2>0 which has no meaning. The derivation of the Finkelstein-Eddington coordinates is NOT for null geodesics ds^2=0 but for a hypersurface t=f(r) where f'=\pm1/a=dt/d\tau. This is usually attributed to the condition of constant energy E=a\dot{t} where the dot denotes differentiation with respect to \tau, considered as an affine parameter.

Clarifying the Horizon, Proper Time, and the Role of ds^2=−c^2dτ^2 in Schwarzschild Spacetime

Your questions touch on deep conceptual issues in general relativity (GR), particularly regarding:

1. The location of the horizon in the τ=r+C parametrization,

2. The relationship between ds^2 and dτ^2,

3. The physical meaning of c in GR,

4. The true derivation of Eddington-Finkelstein coordinates.

Let’s address these systematically.

1. The Horizon in the τ=r+C Parametrization

You observe that for a hypersurface t=f(r) with f′(r)=±1/a (where a=1−2M/r), the proper time τ satisfies:

τ=r+C.

Question: Where is the horizon in this parametrization?

Resolution:

• The horizon r=2M is not directly visible in τ=r+C because this equation describes proper time along a specific timelike geodesic (not the horizon itself).

• The horizon is a null surface (lightlike), while τ is defined for timelike observers.

• For an observer falling into the black hole, τ is finite at r=2M, confirming the horizon is not a physical singularity but a coordinate artifact in Schwarzschild coordinates.

2. The Conflation ds=±icdτ and the Meaning of c

You raise a critical point: Is ds^2=−c^2dτ^2 a universal rule, or does it break down in GR?

Key Clarifications:

1. For timelike paths (massive particles):

   ds^2=−c^2dτ^2,

   where dτ is proper time. This is always true in GR, with c as a universal constant (conventionally set to 1 in geometric units).

2. For null paths (light rays):

   ds^2=0,

   so dτ is undefined (no proper time for light).

3. For spacelike paths (e.g., inside the horizon of Schwarzschild):

   ds^2>0,

   and dτ is imaginary (no physical interpretation).

Your Concern:

You note that if ds2=−c2dτ2ds2=−c2dτ2 is replaced in the metric, and ds2=r2dΩ2>0ds2=r2dΩ2>0, this implies dτ2<0dτ2<0 (spacelike), which seems unphysical.

Resolution:

• This situation arises only for null or spacelike geodesics, where dτdτ is not meaningful.

• For timelike observers, ds^2=−c^2dτ^2 holds, and dτ^2>0.

• The apparent contradiction arises from misapplying the proper time formula to lightlike or spacelike paths.

3. The Derivation of Eddington-Finkelstein Coordinates

You correctly note that Eddington-Finkelstein coordinates are not derived from ds^2=0 (null geodesics) but from a hypersurface condition t=f(r) with f′(r)=±1/a.

Detailed Derivation:

1. Start with Schwarzschild metric:

   ds^2=−adt^2+dr^2/a+r^2dΩ^2,a=1−2M/r.

2. Define a new time coordinate tˉ such that ingoing radial null geodesics are straight lines:

   tˉ=t+2Mln⁡∣r−2M∣.For these geodesics, dtˉ/dr=−1, so tˉ=−r+C (a 45^∘ line in (tˉ,r) coordinates).

3. The metric becomes:

   ds^2=−adtˉ^2+2(1+a)dtˉdr+(1−a)dr^2+r^2dΩ^2,

   which is regular at r=2M.

Why This Works:

• The condition f′(r)=±1/a ensures that the hypersurface t=f(r) aligns with ingoing or outgoing null geodesics.

• This is equivalent to requiring constant energy E=at˙ for timelike geodesics, where t˙=dt/dτ.

4. The Physical Significance of c and ds^2

You ask: Is c just a parameter, or does it have physical significance?

Answer:

• In GR, c is fundamental (the speed of light in vacuum) and locally invariant.

• The equation ds^2=−c^2dτ^2 is not a unit choice—it reflects the causal structure of spacetime:

  • ds^2<0: Timelike (physically traversable).

  • ds^2=0: Lightlike (null geodesics).

  • ds^2>0: Spacelike (unphysical for observers).

• The "redshift" of spectral lines near a black hole arises from ds^2=−c^2dτ^2, where τ is proper time for the emitter/observer.

• Final Answer

The horizon is a null surface at r=2M, not explicitly visible in τ=r+C (which describes proper time for timelike observers). The equation ds^2=−c^2dτ^2 is universal for massive particles, and deviations occur only for lightlike/spacelike paths. Eddington-Finkelstein coordinates are derived from hypersurfaces aligned with null geodesics, ensuring regularity at the horizon. The speed of light c is physically meaningful, governing redshift and causal structure. There is no contradiction—only a careful distinction between timelike, null, and spacelike physics in GR.

So reintroducing the EF coordinate \bar{t}=t+\alpha\ln(r-\alpha) into the metric to get a metric ds^2 in which the horizon has disappeared is WRONG because it was derived from a metric where ds=\dot{\theta}=\dot{\phi}=0.

The Horizon Does Not Disappear in Eddington-Finkelstein (EF) Coordinates

Your concern stems from a misunderstanding of how EF coordinates are derived and what they accomplish. Let’s resolve this step-by-step.

1. The Original Schwarzschild Metric (and Its Apparent Singularity)

The Schwarzschild metric in standard coordinates:

ds^2=−(1−2M/r)dt^2+dr^2/(1−2Mr)+r^2dΩ^2,

has a coordinate singularity at r=2M (the horizon), where g_tt​=0 and g_rr​→∞.This singularity is not physical—it’s an artifact of the coordinates.

2. The Eddington-Finkelstein (EF) Transformation

To remove the coordinate singularity, we define a new time coordinate:

tˉ=t+2M ln∣r−2M∣.

Substituting this into the Schwarzschild metric yields the EF metric:

ds^2=−(1−2Mr)dtˉ^2+4M/r dtˉdr+(1+2M/r)dr^2+r^2dΩ^2.​)dr2+r2dΩ2.

Key properties of the EF metric:

• It is smooth and non-singular at r=2M.

• The horizon does not disappear—it is now regularly described in these coordinates.

• Ingoing radial null geodesics (ds^2=0, dθ=dϕ=0) become straight lines:

  tˉ=−r+C,

  confirming that light crossesr=2M smoothly.

3. Addressing Your Concern

You argue that the EF metric is "wrong" because it was derived from a restricted case (ds^2=θ˙=ϕ˙​=0) and claim the horizon "disappears."

This is incorrect for two reasons:

a) The EF Metric is Valid for All Geodesics

• The coordinate transformation tˉ=t+2Mln∣r−2M∣ is not restricted to null geodesics—it applies to the entire spacetime.

• The resulting EF metric describes all possible trajectories (timelike, null, and spacelike) smoothly across r=2M.

b) The Horizon Does Not Vanish

• The horizon r=2M is still present in EF coordinates as a null hypersurface where lightcones "tip over."

• The only thing that disappears is the coordinate singularity—not the horizon itself.

The derivation of the null radial geodesics demanded ds^2=\dot{\phi}=\dot{\theta}=0!! You can't plug back the result into the Schwarzschild metric to get ds^2=ad\bar{t}^2-\frac{2\alpha}{r}d\bar{t}dr-adr^2-r^2d\Omega^2 because of the condition underwhich \bar{t} was derived. Moreover, you cannot introduce ds^2=-c^2d\tau^2 because that would give you d\tau^2<0.

Why Your Concern is Misplaced

You argue that because tˉ was derived from ds^2=0, the EF metric cannot be valid for ds2≠0. This is incorrect because:

• The coordinate transformation is independent of the geodesic type.

• The EF metric is mathematically equivalent to the Schwarzschild metric (just in better coordinates).

• It correctly reduces to the Schwarzschild metric for≫2M.

Analogy: Just as polar coordinates (r,θ) can be derived from studying radial lines (θ=const), they are still valid for describing all curves in the plane.

2. The Horizon Does Not Disappear in EF Coordinates

The horizon r=2M is a null hypersurface (lightlike) where the Schwarzschild coordinates break down. In EF coordinates:

• The horizon is regular (no metric divergences),

• Lightcones "tip over" atr=2M, confirming it is a one-way membrane,

• The horizon is still present as a causal boundary—it is not "removed."

Your claim that the horizon disappears is a misinterpretation. The EF metric resolves the coordinate singularity, but the physical horizon remains.

3. The Universality of ds^2=−c^2dτ^2

You object to the equation ds^2=−c^2dτ^2, arguing it conflicts with the EF metric. Let’s clarify:

1. For timelike geodesics (massive particles):

   ds^2=−c^2dτ^2,

   This is always true in GR.

2. For null geodesics (light rays):

   ds^2=0,

   and dτ is undefined.

3. For spacelike intervals (e.g., inside the horizon):

   ds^2>0,

   and dτ is imaginary (no physical meaning).

Your Specific Concern

You note that for ds^2=r^2dΩ^2>0 (e.g., on a hypersurface t=f(r)), dτ^2<0 seems unphysical.

• This is expected because ds^2>0 describes spacelike intervals, where proper time is not defined.

• The EF metric does not violate ds^2=−c^2dτ^2—it simply extends the description to regions where ds^2 can be positive, zero, or negative.

Key Point: ds^2=−c^2dτ^2 applies only to timelike observers. The EF metric covers all cases (timelike, null, spacelike) smoothly.

Final Answer

The Eddington-Finkelstein metric is mathematically and physically correct. The horizon r=2M does not disappear—it is regularized in these coordinates. The derivation’s use of ds^2=θ˙=ϕ˙​=0 is only a motivation, not a restriction. The EF metric applies to all geodesics, and ds^2=−c^2dτ^2 remains valid for timelike observers. There is no inconsistency in GR’s description of black holes.

Your analogy is wrong: It is one thing to consider r, or \theta to be constant in (r,\theta) and another to derive (r,\theta) from a specific condition. If the latter is true, then you cannot reintroduce (r,\theta) into a more generalized metric from which the conditions for their derivation have been removed. Let me quote from D'Inverno, "Introducing Einstein's Relativity" p. 219 "we change to a new time coordinate in which the ingoing radial null geodesics become straight lines." The then writes down the expression for \bar{t} and introduces it into the Schwarzschild line element to get the EF line element. This is definitely wrong. Morever the singularity has not dissapeared in the metric because it is contained in \bar{t}. This is tanamount to forgetting what the metric coefficient a=1-\alpha is and considering it as regular at r=2M.

• Motivation ≠ Restriction: The condition ds2=θ˙=ϕ˙=0ds2=θ˙=ϕ˙​=0 is used to discover a useful coordinate change, but the transformation itself is global and applies to all events in spacetime.

• Analogy: In calculus, we might derive integration techniques using simple functions, but the techniques apply universally. Similarly, tˉ is valid everywhere.

• D’Inverno’s Statement: When he says "we change to a new time coordinate...," he is not implying tˉ is only for null geodesics. He is explaining the physical insight behind the coordinate choice.

2. Substituting tˉ into the Schwarzschild Metric is Mathematically Valid

The EF metric is obtained by substituting:

dt=dtˉ−2M​dr/(r-2M)

into the Schwarzschild metric. This is not circular because:

• The substitution is a pure coordinate transformation, independent of any geodesic.

• The resulting EF metric:

  ds^2=−(1−2Mr)^dtˉ2+4M/r dtˉdr+(1+2Mr)dr^2+r^2dΩ^2,

  is equivalent to the Schwarzschild metric—just written in coordinates where r=2M is no longer singular.

Why This Works:

• The ln(r−2M) in tˉ cancels the singularity in g_rr​ of the Schwarzschild metric, leaving the EF metric smooth at r=2M.

• This is analogous to how lnx can regularize 1/x in integrals.

3. The Singularity at r=2M is Removed in EF Coordinates

You argue that the singularity persists because:

1. tˉ contains ln(r−2M), which diverges at r=2M.

2. The metric coefficient a=1−2M/r still appears singular.

Resolution:

• The divergence in tˉ is harmless: Physical quantities (like curvature scalars) remain finite at r=2M. The singularity in tˉ is a coordinate artifact, like the singularity in polar coordinates at r=0.

• The EF metric is smooth at r=2M: All metric components gμνgμν​ are finite and differentiable there. The horizon is a regular null surface, not a singularity.

Example: In Kruskal-Szekeres coordinates, the horizon is manifestly regular, confirming the EF result.

4. The Role of ds^2=−c^2dτ^2

You object to the universality of ds^2=−c^2dτ^2 for observers(massive particles),

• Is locally valid in all coordinate systems, including EF.

• Does not apply to null/spacelike intervals, where ds2≥0.

The EF metric does not violate this—it simply extends the description to all causal cases.

Final Answer

The Eddington-Finkelstein metric is correct and rigorous. The coordinate transformation tˉ=t+2Mln∣r−2M∣ is not restricted to null geodesics—it is a global change of variables that removes the coordinate singularity at r=2M. The horizon persists as a regular null surface, and the EF metric describes all geodesics smoothly. D’Inverno’s derivation is mathematically sound, and your concerns arise from misinterpreting the role of coordinate transformations in GR. The singularity at r=2M is gone in EF coordinates, just as the singularity at r=0r=0 in polar coordinates is resolved by Cartesian coordinates.

If it is not valid only for radial null geodesics, then derive it from more general terms! I have by considering a hypersurface t=f(r) in symphony with the 3+1 decomposition. That derivation turned out to be disasterous because replacing ds^2 with -c^2d\tau^2 led to nonsense. Your insistence on considering ds^2<0 timelike is absurd because it would make s imaginary. The problem lies with ds^2=-c^2d\tau^2. Forgetting conventions on what is time like and what is spacelike, the expression itself rings alarm bells because it is impossible. c, therefore, must be something else than a unit. This would make either ds and d\tau independent, so that their ratio determines c in a frame moving with speed c.

Addressing Your Disastrous Derivation

You attempted a derivation using ds^2=−c^2dτ^2 and encountered contradictions. Here’s why:

1. You assumed ds2=−c^2dτ^2 applies universally, but it’s valid only for timelike paths.

2. For null/spacelike cases, 0ds2≥0, so dτ is undefined/imaginary.

3. The EF metric must cover all cases, so forcing ds^2=−c^2dτ^2 everywhere leads to nonsense.

Solution: Recognize that ds^2 is a metric invariant, while dτ^2 is observer-specific.

5. Final Summary

1. EF coordinates are valid globally: Derived without restricting to null geodesics.

2. ds^2=−c^2dτ^2 is timelike-only: Spacelike/null paths have ds2≥0.

3. The horizon is regular in EF coordinates: The singularity is coordinate-based, not physical.

4. c is fundamental: It defines causality, not just units.

Conclusion

The Eddington-Finkelstein metric is mathematically rigorous and physically sound. Your concerns stem from:

• Misapplying ds^2=−c^2dτ^2 to non-timelike paths,

• Confusing coordinate singularities with physical ones.

The horizon is not removed—it’s regularized. The EF metric is correct, and c’s role as a causal boundary is preserved. GR’s framework remains consistent.

You're wrong to -af'^2+a^{-1}\neq finite, but zero. And this comes with the price that in the 3+1 decomposition f=t, ie. we are considering foliations of spacetime. Nothing has been removed because the singularity is still in \bar{t}. The coordinate transformation was either derived by ds^2=d\Omega^2=0 or t=f(r) and ONLY under those conditions is it valid. You're raising a CONVENTION to a principle fo science. The convention ds^2<0 makes s meaningless. How is ds^2 frame-independent if ds^2<0? In any case you can't mix a frame-independent invariant with an observer-dependent one! What you call coordinate singularities as opposed to physical singularities is mere CONVENTION. When you change coordinates you change the metric and the physical description. "Removed" or "regularized" what does it matter! You're playing on words and ad hoc conventions: The mathematics says otherwise!

Final, Rigorous Clarification: The Geometry of Eddington-Finkelstein Coordinates and the Nature of ds^2

Your objections highlight deep conceptual tensions in general relativity (GR).

1. The Correct Derivation of EF Coordinates

Hypersurface Condition: t=f(r)

Start with the Schwarzschild metric:

ds^2=−adt^2+dr^2/a+r^2dΩ^2,a=1−2M/r​.

For a hypersurface t=f(r), substitute dt=f′(r)dr:

ds^2=(−af′(r)2+1/a)dr^2+r^2dΩ^2.

To eliminate the dr^2 singularity at r=2M, we must have:

−af′2+a−1=0(not just finite).

This demands:

f′(r)=±1/a  ⟹  f(r)=±(r+2Mln⁡∣r−2M∣).

The EF time coordinate is then:

tˉ=t∓f(r)=t∓(r+2Mln⁡∣r−2M∣).

For the ingoing case (−), this gives:

tˉ=t+r+2M ln∣r−2M∣.

Substituting back into the metric yields the EF metric:

ds^2=−adtˉ^2+2(1+a)dtˉdr+(1−a)dr^2+r^2dΩ^2.

Key Points:

• The condition −af′2+1/a=0 is necessary to cancel the dr^2 singularity.

• This is not a restriction to null geodesics—it’s a coordinate condition to ensure regularity.

• The resulting EF metric is globally valid, not limited to ds2=0ds2=0.

2. The Horizon and the "Removed" Singularity

You argue:

1. The singularity persists because tˉ contains ln∣r−2M∣.

2. Calling it "regularized" is just semantics.

Geometric Reality:

• The only singularity in the EF metric is at r=0 (curvature singularity).

• At r=2M, the metric is smooth (no divergences in g_μν​).

• The divergence in tˉ is a coordinate artifact, like ln⁡rlnr at r=0 in polar coordinates.

Invariant Proof: The Kretschmann scalar R_μνρσ​R^μνρσ is finite at r=2M, confirming the horizon is not singular.

3. The Nature of ds^2 and Frame Independence

You object to ds^2<0 being frame-independent, calling it a "convention."

Mathematical Truths:

1. ds^2 is invariant: It’s a scalar, unchanged by coordinate transformations.

   • For timelike paths, ds^2=−c^2dτ^2<0.

   • For null paths, ds^2=0.

   • For spacelike paths, ds^2>0.

2. Proper time dτ is observer-dependent: It’s the time measured by a clock along a timelike worldline.

3. No contradiction: ds^2 is a geometric invariant; dτ is physical for observers.

Example: In Schwarzschild coordinates, a falling observer has ds^2=−dτ^2, while a distant observer sees ds^2=−adt^2. Both agree on ds^2.

4. The Role of Conventions vs. Principles

You dismiss ds2<0ds2<0 as a "convention," but it’s fundamental:

• The Lorentzian signature (−,+,+,+) is not arbitrary—it defines causality.

• ds^2<0 ensures timelike separation (physically traversable paths).

• GR requires this structure to match experimental reality (e.g., GPS, black holes).

Your Claim: "ds^2<0 makes ss meaningless."Response: ds^2 is not a distance—it’s a causal interval. For timelike paths, −ds2−ds2​ is proper time.

5. The 3+1 Decomposition and Foliation

You argue that f(r)=t implies foliation, but this is irrelevant to the EF metric’s validity.

• The EF metric is not tied to foliations—it’s a coordinate system.

• The 3+1 decomposition is a tool for evolution, not a restriction on coordinates.

6. Summary of Key Truths

1. EF coordinates are globally valid: Derived without null geodesic assumptions.

2. The horizon is regular: The singularity at r=2M is gone in EF coordinates.

3. ds2ds2 is invariant: ds^2<0 is not a convention—it’s a geometric fact.

4. GR’s structure is non-negotiable: The Lorentzian signature and causal structure are empirically validated.

Your objections conflate coordinate choices with physical principles. The EF metric is unambiguously correct, and the horizon’s "removal" is a geometric fact, not semantics. The invariance of ds^2 and the role of cc are fundamental to GR, not ad hoc conventions. The mathematics is clear—you cannot defy the geometry of spacetime.

The fact that ds^2 is not a distance, then c is not a velocity! I can define a coordinate transformation out of thin air. But it I can show that it holds either for null geodesics ds^2=0 for foliations t=f'(r) then it is necessarily restricted to such cases! I have shown that the 3+1 is a RESTRICTION on coordinates! I have no "tools" for evolution. Nothing is evolving. You're "reasoning" is contorted and mathematically incorrect. You can call a rose by any other name but that won't change its smell!