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The Kruskal-Szekeres coordinates for Schwarzschild spacetime $(T,X,\theta,\phi)$ can be defined as is done in Wikipedia.

Now, what is the physical meaning of these coordinates? The standard Schwarzschild coordinates $(t,r,\theta,\phi)$ can be physicaly interpreted as the coordinates of events as seen by an observer at infinity.

What about the KS coordinates? In the book Introduction to Quantum Effects in Classical Backgrounds by V. F. Mukhanov and S. Winitzki, in section 9.1.2 they explain:

It is a standard result that the singularity in the Schwarzschild metric which occurs at $r = 2M$ is merely a coordinate singularity since a suitable change of coordinates yields a metric regular at the BH horizon. For instance, an observer freely falling into the black hole would see a normal, finitely curved space while crossing the horizon line $ r= 2M$. Therefore one is motivated to consider a coordinate system $(\bar{t},\bar{r})$ that describes the proper time $\bar{t}$ and the proper distance $\bar{r}$ measured by a freely falling observer. A suitable coordinate system is the Kruskal frame.

Well, this wording makes it seem that $T,X$ have the meaning of "the proper time" and "the proper distance" measured by a freely falling observer. Is that correct?

If so, how can we see this? The formulas for $T,X$ are not intuitive at all, and they clearly don't make this interpretation obvious.

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I (personally) don't think we should be talking about the meaning of the coordinates. The Kruksal metric is a useful re-coordinization of the Schwarzschild spacetime. The 2D diagram for the coordinates. Note that $(\phi ,\theta)$ don't really matter since the system is spherically symmetric.

If you learned about the Schwarzschild metric then you probably saw that the light cones become deformed as you approach the event horizon (and flip once inside), this is because even light finds it hard to leave the vicinity of the black hole due to the warping of spacetime there (not quite gravitational attraction, since gravity as a force doesn't exist in GR). Also, note that the light cones approach those of the Minkowski metric for $r>>2M$.

One of the main differences between the Schwarzschild and the Kruksal metric is that in the latter the light cones are preserved as if it were Minkowski spacetime. The $X$ coordinate lines are hyperbolas instead of vertical lines; whereas the $T$ coordinate lines are straight lines through the origin ($T=0$ is a horizontal line and as $T \rightarrow \infty$ the line approaches the event horizon). The advantage of the Kruksal coordinates is that you can see a lot of pretty things in there:

-A region from which nothing can escape (black hole)

-A region where nothing can remain (white hole, although these are probable just an extra [like the solution of a parabolic trayectory gives two solutions but only the one in the future is relevant])

-Another section of spacetime that is completely isolated from the other (parallel universe)

-A "bridge" between the two universes (at $T=0$ you could cross to the other side... if you could go faster than light... so... nope) (wormhole)

There is another conformal transformation used for Penrose diagrams that features the same things but its nicer in some ways. (Just google "Penrose diagram Schwarzschild").

Hope it helped.

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The physical meaning of the Kruskal–Szekeres coordinates is that both infalling and outgoing null geodesics correspond to 45° inclined lines on the spacetime diagram.

More precisely, radial null geodesics correspond to $$\begin{array}{l} T+X=\mathrm{const},\qquad \text{infalling,}\\ T-X=\mathrm{const},\qquad \text{outgoing.} \end{array} $$

As to the other part of the question:

it seem that $T$, $X$ have the meaning of "the proper time" and "the proper distance" measured by a freely falling observer. Is that correct?

Since line element in terms of $T$, $X$ has a factor in front of $dT^2$ that is diverging as singularity ($r=0$) is approached, the answer to that question seems to be 'no'.

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I am not sure if your understanding about physical meaning of Schwarzscild coordinates is correct. In GR, I often heard that the local quantities do not make sense because of diff. invariance.

Schwarzscild geometry can be understood using the following picture: suppose there is a observer at each position of space with fixed value $(r,\theta,\phi)$ (called Schwarzschild observer), each observer carries two clocks. One is the standard clock which record his proper time. One Schwarzscild clock records the coordinate time t which will runs at different proper rate at different position. At infinity, the two clocks run at the same rate. At each position, we have $$\mathrm{d}\tau~=~\sqrt{1-\frac{2GM}{r}} \, \mathrm{d}t~=~\left.\mathrm{d}t\right|_{t \to \infty}\,.$$ The proper distance of Schwarzschild observer is $$\mathrm{d}l~=~\frac{\mathrm{d}r}{\sqrt{1-\frac{2GM}{r}}}\,.$$

For the Kruskal coordinates which is good for free falling observer since it covers the whole geometry $$\mathrm{d}s^2~=~-\frac{32G^3M^3}{r} \, \exp{\left(-\frac{r}{2GM}\right)} \left(\mathrm{d}T^2-{\mathrm{d}Z}^2 \right)+r^2 \,\mathrm{d}\Omega^2$$ For the radial free falling observer, at each position, the proper time is $\mathrm{d}\tau \propto \mathrm{d}T$, and the proper distance is $\mathrm{d}l \propto \mathrm{d}Z$.

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