If, in the FRW metric, the 'interval' is:

 

[math]ds^2 = (c \, dt)^2 - R(t)^2 \left( \frac{dr^2}{1 - K r^2} + r^2 \left( d \theta^2 + ( sin(\theta) d\phi )^2 \right) \right)[/math]

And if, in GR, that 'interval' is derived from the 'metric' tensor, via:

 

[math]ds^2 = g_{\mu \nu} ds^{\mu} ds^{\nu}[/math]

And if [math]ds \equiv \left[ c dt, dr, d\theta, d\phi \right][/math]; then, why wouldn't the 'metric' tensor be:

 

[math] \left( \begin{array}{cccc}

1 & 0 & 0 & 0 \\

0 & R^2 / 1- K r^2 & 0 & 0 \\

0 & 0 & R^2 r^2 & 0 \\

0 & 0 & 0 & R^2 r^2 sin(\theta)^2 \end{array} \right)[/math]

or, if [math]ds \equiv \left[ c dt, R dr, R r d\theta, R r sin(\theta) d\phi \right][/math] (to put all the physical distances into the interval); then, why wouldn't the 'metric' tensor be:

 

[math] \left( \begin{array}{cccc}

1 & 0 & 0 & 0 \\

0 & 1 / 1- K r^2 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \end{array} \right)[/math]

?

Are you asking about the minus sign difference between the time and space?

According to Eric V. Linder's First Principles of Cosmology, pg.88,

 

[math]1+z = \frac{\left( \vec{k} \circ \vec{u} \right)_e}{\left( \vec{k} \circ \vec{u} \right)_o}[/math]

where [math]\vec{k} \circ \vec{u} = g_{\mu \nu} k^{\mu} u^{\nu}[/math]. Yet, when that four-dot-product is taken, the Minkowski (locally flat-space) metric tensor seems to be utilized, such that (for a radially-inbound [math]e \rightarrow o[/math] photon):

 

[math]1+z = \frac{\lambda_o}{\lambda_e} \frac{\gamma_e \left( 1 + \beta_{r,e} \right)}{\gamma_o \left( 1 - \beta_{r,o} \right)}[/math]

What would happen, if you considered cosmologically-distant light-sources, deep down in a gravity well? If, as a first approximation, one were to assume a Schwarzschild space-time, is not the Schwarzschild Metric:

 

[math] \left( \begin{array}{cccc}

1-R_S/r & 0 & 0 & 0 \\

0 & -1 / (1- R_S/r) & 0 & 0 \\

0 & 0 & -1 & 0 \\

0 & 0 & 0 & -1 \end{array} \right)[/math]

? Then, the four-dot-product time-time components [math]1 \rightarrow 1-R_S/r[/math]; and, the space-space components [math]\beta_r \rightarrow \beta_r / (1-R_S/r)[/math], so that:

 

[math]1+z = \frac{\lambda_o}{\lambda_e} \frac{\gamma_e \left( (1- \frac{R_{S,e}}{r}) + \frac{\beta_{r,e}}{1-R_{S,e}/r} \right)}{\gamma_o \left( (1- \frac{R_{S,o}}{r}) - \frac{\beta_{r,o}}{1-R_{S,o}/r} \right)} \approx \frac{\lambda_o}{\lambda_e} \frac{ \left( (1- \frac{R_{S,e}}{r}) + \beta_{r,e} (1+R_{S,e}/r) \right)}{\left( (1- \frac{R_{S,o}}{r}) - \beta_{r,o} (1+R_{S,o}/r) \right)} = \frac{\lambda_o}{\lambda_e} \frac{ \left( 1 + \beta_{r,e} - \frac{R_{S,e}}{r} (1 - \beta_{r,e}) \right)}{\left( 1 - \beta_{r,o} - \frac{R_{S,o}}{r} (1 + \beta_{r,o}) \right)}[/math]

to lowest orders, in velocities and radii terms. For the Schwarzschild terms, the time-originating piece dominates the space-originating piece, and actually reduces the redshift, if the emitter is deep down in a gravity well. Could that be correct ?? Is not the R_S of our galaxy, of [math]\approx 10^{12} M_{\odot}[/math], thusly about one-third-light-year ? If so, then for us, out ~30Kly from our galactic core, the effect would be O(10^-5).

 

EDIT: To emphasize the 'gravity-well' effect, assume zero velocity; and, assume that some emission line implies [math]\lambda_e[/math], whilst [math]\lambda_o[/math] is known. Then:

 

[math]\frac{1 - \frac{R_{S,o}}{r} }{ 1 - \frac{R_{S,e}}{r}} (1+z) \approx\frac{\lambda_o}{\lambda_e} [/math]

showing that light, blue-shifting in-bound at the observer, much increase the inferred red-shift; and, to account for non-cosmological red-shifting out-bound at the emitter, "before the light is loosed-and-let-to-fly out into the Hubble Flow", one must reduce their inferred cosmological red-shift value. Could this be relevant for photons emitted near compact objects, e.g., GRBs ??

 

However, for GRBs, I understand that [math]\lambda_e[/math] is unknown; and, that known cosmological red-shifts are obtained, separately, from their host galaxy. Thus,

 

[math] \lambda_e \approx \frac{\lambda_o}{1+z} \frac{1 - \frac{R_{S,e}}{r} }{ 1 - \frac{R_{S,o}}{r}}[/math]

showing that as [math]r_e \rightarrow R_{S,e}[/math], then [math]\lambda_e \rightarrow 0[/math], i.e., the radiation is "harder than it looks". And, indeed, long-period GRBs typically are observed to have softer radiation, but be BH-generated; whereas short-period "short hard" GRBs typically are observed to have harder radiation, but may be NS-merger generated. That picture appears to be completely self-consistent, since BH-generated l-GRBs emit their light closer to the Schwarzschild radius.

Edited August 26, 201114 yr by Widdekind

re-EDIT: To emphasize the 'gravity-well' effect, assume zero velocity; and, assume that some emission line implies [math]\lambda_e[/math], whilst [math]\lambda_o[/math] is known. Then:

 

[math]\frac{1 - \frac{R_{S,o}}{r} }{ 1 - \frac{R_{S,e}}{r}} (1+z) \approx\frac{\lambda_o}{\lambda_e} [/math]

showing that light, blue-shifting in-bound at the observer, much increase the inferred red-shift; and, to account for non-cosmological red-shifting out-bound at the emitter, "before the light is loosed-and-let-to-fly out into the Hubble Flow", one must reduce their inferred cosmological red-shift value. Could this be relevant for photons emitted near compact objects, e.g., GRBs ??

 

However, for GRBs, I understand that [math]\lambda_e[/math] is unknown; and, that known cosmological red-shifts are obtained, separately, from their host galaxy. Thus,

 

[math] \lambda_e \approx \left( \frac{\lambda_o}{1+z} \right) \frac{1 - \frac{R_{S,e}}{r} }{ 1 - \frac{R_{S,o}}{r}}[/math]

showing that as [math]r_e \rightarrow R_{S,e}[/math], then [math]\lambda_e \rightarrow 0[/math], i.e., the radiation is "harder than it looks". And, indeed, long-period GRBs typically are observed to have softer radiation (~1/3 m_e c²), but be BH-generated; whereas, short-period "short hard" GRBs typically are observed to have harder radiation (~1/2 m_e c²), but may be NS-merger generated (Mazure & Basa. Exploding Superstars; Bloom. What are GRBs ?). That picture appears to be completely self-consistent, since BH-generated l-GRBs emit their light closer to the Schwarzschild radius.

 

Note, tho, that equally plausibly, humans on earth only observe GRBs, when looking "down the bore-sight", straight towards a pole, of the central compact object. Thus, matter in-falling towards said central accretor, would be moving radially away from earth -- and, hence, any light emitted "back up-gravity-well", towards earth, would be redshifted. Thus, matter infalling faster, towards a BH, during an l-GRB, could also account for the inferred redshifting of the radiation (assumed to be from pair production, at 511 KeV), relative to a somewhat slower infall, towards the NSs, in an s-GRB. Therefore, matter infalling towards the central compact object, at a considerable fraction of the speed-of-light; or, radiating matter residing, close to said compact object (~1.5-2 R_S), could both account, jointly or severally, for the relative redshift, of l-GRBs, compared to s-GRBs.

Edited August 27, 201114 yr by Widdekind

According to Eric V. Linder's First Principles of Cosmology, pg.88,

 

[math]1+z = \frac{\left( \vec{k} \circ \vec{u} \right)_e}{\left( \vec{k} \circ \vec{u} \right)_o}[/math]

where [math]\vec{k} \circ \vec{u} = g_{\mu \nu} k^{\mu} u^{\nu}[/math]...

 

What would happen, if you considered cosmologically-distant light-sources, deep down in a rotating gravity well? If, as a first approximation, one were to assume a Kerr space-time, then would -- for calculating the 4-dot product at the emitter -- not the Schwarzschild Metric, for a polar-emitted photon ([math]\theta = d\theta = d\phi \equiv 0[/math]), e.g., long-type-GRBs, look vaguely like...

 

[math] \left( \begin{array}{cccc}

1- r R_S/(r^2 + \alpha^2) & 0 & 0 & 0 \\

0 & -(r^2 + \alpha^2) / (r^2 - \alpha^2 + r R_S) & 0 & 0 \\

0 & 0 & -1 & 0 \\

0 & 0 & 0 & -1 \end{array} \right)[/math]

? At the emitter, the photon 4-vector is, in natural units (h = c = 1), [math]\lambda_e^{-1} \left( \begin{array}{c}

1 \\

-1 \\

0 \\

0 \end{array} \right)[/math]; and, the emitter 4-vector is [math]\gamma_e \left( \begin{array}{c}

1 \\

\beta_{e,r} \\

\beta_{e,\theta} \\

\beta_{e,\phi} \end{array} \right)[/math]. And then, the four-dot-product, at the emitter, becomes:

 

[math]\left( \vec{k} \circ \vec{u} \right)_e = \left( \frac{\gamma_e}{\lambda_e} \right) \left( \left[ 1- \frac{r R_{S,e}}{r^2 + \alpha^2} \right] + \beta_{e,r} \left[ \frac{r^2 + \alpha^2}{r^2 - \alpha^2 + r R_{S,e}} \right] \right)[/math]

And more, borrowing from the PP, the four-dot-product, at the observer, assumed-to-be in the observer's rest-frame, where [math]\gamma_o = \beta_o = 0[/math]; and, which rest-frame is further assumed-to-be non-rotating; thusly becomes:

 

[math]\left( \vec{k} \circ \vec{u} \right)_o = \left( \frac{1}{\lambda_o} \right) \left( 1- \frac{R_{S,o}}{r} \right)[/math]

And then:

 

[math]1+z = \frac{\left( \vec{k} \circ \vec{u} \right)_e}{\left( \vec{k} \circ \vec{u} \right)_o}[/math]

 

[math]\; \; \; \; = \left( \frac{\lambda_o}{\lambda_e} \right) \left( \frac{\gamma_e}{1- \frac{R_{S,o}}{r}} \right) \times \left( \left[ 1- \frac{r R_{S,e}}{r^2 + \alpha^2} \right] + \beta_{e,r} \left[ \frac{r^2 + \alpha^2}{r^2 - \alpha^2 + r R_{S,e}} \right] \right) [/math]

For a maximally-rotating Kerr BH, the "Kerr radius" [math]\alpha = R_S / 2[/math]; and, emission can occur, from this "rotationally reduced" radius [math]r = \alpha[/math]. Assuming such, the above formula simplifies to:

 

[math]1+z = \left( \frac{\lambda_o}{\lambda_e} \right) \left( \frac{\gamma_e}{1- \frac{R_{S,o}}{r}} \right) \times \beta_{e,r} [/math]

Ignoring the O(10^-5) effect, of our own assumed-to-be-the-observer gravity well, w.h.t.:

 

[math]1+z = \left( \frac{\lambda_o}{\lambda_e} \right) \times \left( \gamma_e \, \beta_{e,r} \right) \rightarrow \infty [/math]

What is the formula, for the "fall-back radius", for a given (rotating) BH, and a given "initial launch velocity", i.e., given [math]\beta_r[/math], how close can you get to a given (rotating) BH, and still escape ?

 

Assuming that, per PP, [math]\lambda_o \approx 1/3 \lambda_e[/math]; and, assuming an average red-shift for long-GRBs of [math]<z> \approx 0.7[/math] (Mazure. Exploding Super-Stars); and, assuming essentially totally-radial (electron) emitter velocity; then, [math]\gamma_e \beta_e \approx 5[/math], i.e., ~0.98 c. Presumed-to-have-fallen-back electrons, acted, "long long ago, far far away", like "launch-vehicle rocket sleds for barely-escaping gamma-ray photons" ??

 

 

 

 

ADDENDA: This source seems partially poignant, and particularly interesting.