[geordief (OP)]

....or any similar object that curves spacetime?

ie how does the r/t ratio in the s^2= ct^2-r^2  formula change?

Edit :I don't mean the same event ,of course .I mean  identical events whose spacetime intervals are measured as they approach the source of gravity.(Perhaps  light reflecting between two mirrors would do - or two hypothetical events involving massive objects)

Reedit: the event pairs  are causally connected.

[Halc]

What happens to the spacetime interval between two events as they approach a BH?

Events are points in spacetime and as such, do not approach anything. Your first edit seems to indicate your awareness of this.

A pair of events anywhere can have time-like separation, light-like separation, or space-like separation, corresponding respectively to positive, zero and negative intervals.

ie how does the r/t ratio in the s^2= ct^2-r^2  formula change?

That shouldn't change at all. It's frame invariant, so it is the same in any coordinate system of choice.

Reedit: the event pairs  are causally connected.

OK, then their interval is non-negative, and is most simply expressed as zero (if the connection is at light speed) or as the time as measured by a clock following an inertial worldline connecting the two events.

[geordief (OP)]

That shouldn't change at all. It's frame invariant, so it is the same in any coordinate system of choice.

I am talking about similar pairs of events ( say a ball striking a surface ,event#1  and rebounding and striking another surface I metre distant,event#2)

From the frame of a distant observer there is a spacetime interval between those two events.

If we repeat the experiment closer to BH  then I am asking how the  new spacetime interval compares to the earlier spacetime interval ( and how the r/t ratio developes as the experiments are made closer to the BH(or any gravitational source)

[Halc]

It's frame invariant, so it is the same in any coordinate system of choice.

I am talking about similar pairs of events ( say a ball striking a surface ,event#1  and rebounding and striking another surface I metre distant,event#2)

From the frame of a distant observer there is a spacetime interval between those two events.

Again, the interval is frame invariant, so it is the same for all observers.  In this case it is some positive value corresponding to the square of the elapsed time as measured by the free-falling ball between strikes.

If we repeat the experiment closer to BH

Then they're a different pair of events.

then I am asking how the  new spacetime interval compares to the earlier spacetime interval ( and how the r/t ratio developes as the experiments are made closer to the BH(or any gravitational source)

Depends on how long it is between strikes as measured by the ball.  If it's the same time, then the interval is the same.  The two events could be on either side of the event horizon if you please. It really doesn't make a difference, except in the latter case, the outside observer isn't going to observe both events.

[geordief (OP)]

Then they're a different pair of events.

Yes ,I  appreciate(d) that.

[geordief (OP)]

Depends on how long it is between strikes as measured by the ball.  If it's the same time, then the interval is the same.  The two events could be on either side of the event horizon if you please. It really doesn't make a difference, except in the latter case, the outside observer isn't going to observe both events.

If the spacetime interval doesn't change does the spatial component change wrt the temporal component?

[Halc]

If the spacetime interval doesn't change does the spatial component change wrt the temporal component?

The spatial and temporal coordinates of any pair of events is coordinate system dependent, so yes, both change with a change in coordinate system (observer frame if you will).

[geordief (OP)]

The spatial and temporal coordinates of any pair of events is coordinate system dependent, so yes, both change with a change in coordinate system (observer frame if you will).

So ,in my scenario as similar (=identical in their own frame ) pairs of events are recorded by an outside observer  does the ratio of the temporal to the spatial component of the (same) spacetime interval change as the  subsequent event pairs are observed nearer the source of gravity ?

And ,if so which component increases wrt to the other?

Since ,I have heard the time slows from the outside observer's perspective as object's approach an event horizon can I guess it to be the temporal component that increases wrt the spatial component?

[Halc]

So ,in my scenario as similar (=identical in their own frame ) pairs of events are recorded by an outside observer  does the ratio of the temporal to the spatial component of the (same) spacetime interval change as the  subsequent event pairs are observed nearer the source of gravity ?

And ,if so which component increases wrt to the other?

Well in the frame of the ball, there is no spatial component since both events happen at the same location (where the ball is), so it is pure temporal. In any other frame, both values increase, making it more time and nonzero separation in space, so the ratio of temporal to spatial is going to go from infinite (T / 0) to something finite (>T / >0).  This is all true anywhere, so no mention of the black hole is needed.

Since ,I have heard the time slows from the outside observer's perspective as object's approach an event horizon can I guess it to be the temporal component that increases wrt the spatial component?

For the interval to remain invariant, both are going to need to increase. That's simple algebra.

[geordief (OP)]

the ratio of temporal to spatial is going to go from infinite (T / 0) to something finite (>T / >0).  This is all true anywhere, so no mention of the black hole is needed.

So it makes no difference whether the ball is in a region of flat space or curved space?

What about if there is a region of curved space between the observer and the ball ?(which is still falling into its own gravity well.

[captcass]

Again, the interval is frame invariant

Not to the distant observer when the events occur near a BH. A BH is actually a Magnetaspheric Eternally Collapsing Object, or MECO. The center of it is empty space. Rudolph Schild (Harvard/Smithsonian) led a team in 2006 that showed that.

At the event horizon of the MECO, our "limit of relativity", where timelike transitions to spacelike, the distant observer sees that time appears to stop. Of course it is not possible to get to a place where time stops. We are looking down a steep dilation gradient that gets to the point where the difference in the rates of time is 1 s/s. But if one could approach that horizon, it would recede into the empty space inside the MECO. (The same is true at the cosmological horizon, where the Hubble shift makes it appear that time stops as objects appear to accelerate away at c. When approached, it recedes. Another limit of relativity.)

So what the distant observer sees is that the closer the object approaches the MECO, the slower its rate of time so the slower it goes. Time appears to eventually slow to a stop, and all motion ceases, at the event horizon, which would take an eternity to reach.

The object, however, experiences invariant time, like all of us, as it travels down the gradient and the gradient shifts for it as it travels into the space opening ahead of it in the MECO.

This means the galaxies are the branching of the universe into infinity.

Relativity is based upon the fact that, although we know the universe has an invariant rate of time, we cannot see it that way because it takes time for light to travel, creating Lorentz transformations, and energy density slows the rate of time. We see the illusion, not the reality. We see objects moving "through" space, but we know they are densities evolving forward in the quantum continuum. Although we see dimentionality, all three dimensions are being evolved forward simultaneously by time, in the forward direction of time. As time has no depth, neither do the dimensional spatial events being evolved forward. The evolution appears to turn this way and that way towards densities because time slows in densities. The denser the "mass", the greater its resistance to evolution...

So we are not seeing stellar systems revolving around MECOs "through space". We are seeing spatial densities evolving forward at different rates within the evolving continuum.  We see stellar systems evolving forward faster the farther they are from the steep dilation gradient at the center as the continuum evolves forward.

It is doubtful life forms could ever even get close to a MECO due to star density near the center of the galaxy.

Sorry, I wandered a bit there...

[Halc]

So it makes no difference whether the ball is in a region of flat space or curved space?

Exactly

What about if there is a region of curved space between the observer and the ball ?(which is still falling into its own gravity well.

Yet again, it is frame invariant, and thus not observer dependent.

[geordief (OP)]

So it makes no difference whether the ball is in a region of flat space or curved space?

Exactly

What about if there is a region of curved space between the observer and the ball ?(which is still falling into its own gravity well.

Yet again, it is frame invariant, and thus not observer dependent.

Digressing a bit.Is it fair or noteworthy  to say all these events in GR are treated as point objects even though they are actually spatio-temporally extensive?

Is that where a theory of  quantum gravity might be useful?

(I thought it was interesting where you said, to hopefully paraphrase- that the ball was in its own frame of reference and so any interval was purely temporal. )

 

[Halc]

Is it fair or noteworthy  to say all these events in GR are treated as point objects even though they are actually spatio-temporally extensive?

Events are by definition mathematical points in spacetime, say the event of a photon being emitted and such.  In practice, yes, they're spread out, so one can speak of the event of the sinking of the Titanic even though the ship is fairly large and moved around quite a distance during the hours-long process. It just means we lose precision when we talk about that event.
Billiard balls are modeled as point events (the event of the contact between two balls) when in fact the contact takes finite time, involves finite (but very large) acceleration, and finite surface area of contact.

Is that where a theory of  quantum gravity might be useful?

It's just geometry. No, quantum gravity solves other issues.