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White holes, if they exist (unlikely but still fun to think about), are regions of space that can't be entered.
They are time-reversals of black holes (GR is time-symmetric). However, a time reversal of gravitational attraction is still gravitational attraction. This leaves us with a paradox: a region of spacetime that is both attractive and un-enterable.
In light of this conflict, what would happen to an observer falling toward a white hole from rest at a large distance?
Short answer: Fall of an observer toward white hole would convert this white hole into a black hole.
Longer answer. Let us consider a nonrotating white hole that is not currently radiating. The metric outside of this white hole would be the Schwarzschild solution, right until the moment that white hole “explodes”, after which there would be a cloud of “explosion debris” (which could be anything: an expanding cloud of gas, a cloud that would later collapse into a black hole, some radiation and a white hole of smaller mass, a pair of white holes etc.). An inertial observer falling toward this white hole would be moving along a geodesic of Schwarzschild metric. And since by the clock of a far away stationary observer the falling observer would never cross the surface of antihorizon $r=2M$ (in units $G=c=1$), this falling observer would remain outside the white hole until the moment when white hole explodes. After that the falling observer would either be incinerated by the explosion or (if she survived) would continue moving in around explosion debris.
The above description is only possible if we completely disregard the mass of the falling observer. If this observer instead has a mass $m$ (which we consider very small in comparison with the mass $M$ of the white hole), then when the falling observer has radial coordinate $r=2(M+m)$ an event horizon would form enveloping both observer and the white hole. The falling observer and the white hole merge, creating a black hole with mass $M+m$. Timescale for emergence of event horizon is essentially time (by outside clock) needed for an observer to reach $r=2(M+m)$ since the moment of crossing some characteristic surface outside the white hole, say a photon sphere $r=3M$: $$ \tau \approx 2 M \,\ln \frac{M}{m}. $$
This conversion of a white hole into black would happen for any form of mass–energy approaching the antihorizon, no matter how small it is. For example, a typical CMB photon at $\approx 3\,\text{K}$ would convert a white hole with the mass of $10^6\,M_\odot$ in a matter of days after crossing the photon sphere. So a white hole of stellar mass or larger cannot survive for any significant periods of time in the present universe: white holes are unstable against conversion into black holes.
This instability was discovered by Eardley in the 70s:
This white to black conversion scenario has a loophole: white hole could be continuously radiating. Then particles falling toward white hole could be deflected (radiation pressure) and white hole antihorizon could be receding fast enough that no mass–energy would approach it close enough to convert it into the black hole.
In general, there is no gravitational attraction. GR defines no such concept - at least not in general. You have curved spacetime on which test particles move on timelike geodesics ("straight lines in 4D spacetime that go from past to future") and GR will tell you how is the spacetime curved. Sometimes, gravitation can be even repulsive as is the case near a black hole singularity of charged black hole, where there is repulsion proportional to charge of the black hole squared that acts on the neutral particles also (so we cannot interpret the repulsion as due to electromagnetic force and must admit, that repulsion is due to gravitational interaction)
Moreover. The point of white holes - at least those i came across studying simple black holes - is that they live in the past. You can think about them as the singularities from which universe started. Therefore, it makes perfect sense you cannot fall into them, since for that you would need to go back in time. But if i am wrong in this case and there are different white holes that do not offer themselves for this interpretation, i would be happy if (the reader) could let me know in the comment.
