ABSTRACT (original article): The fate of an astronaut unfortunate -- or foolish -- enough to find themselves hurtling towards spaghettification after passing the event horizon of a black hole is a common anecdote told by scientists to the regular population. However, despite the fact the Schwarzschild spacetime has been discovered over a century ago, the simple question of how long can such a space traveller live has not been fully elaborated on since. In fact, a few textbooks even give a mistaken or easily misread description of what happens. We address those inconsistencies. We calculate the proper time a space traveller equipped with means of propulsion can expect to live in these circumstances, giving analytical expressions (as elliptic integrals) wherever possible. We prove a principle that explains the best strategy to extend their life, and show its’ generalisation for other spacetimes. Finally, we give quantitative answers to what gains due to optimal control can be expected in typical and somewhat `realistic’ circumstances. CITATION (original article): Karol Urbański, Prolonging The Inevitable: Maximising survival time of an engine-equipped spacecraft between spatial hypersurfaces, as applied to the Schwarzschild spacetime, arXiv:2405.03510. https://doi.org/10.48550/arXiv.2405.03510
Introduction
Introduction
Recently, I worked on correcting a slight misconception that pops out in a few relativity textbooks. The claim is that were a person to fall into a Schwarzschild black hole, they would survive the longest before they hit the singularity of the black hole by not doing anything, since geodesics (paths where no acceleration is applied by the observer) maximise proper time. Proper time is the time measured by the falling observer on a watch they carry. However, this explanation turns out to be misleading, since the singularity is not a single event, but a hypersurface and so the principle doesn’t apply. There is an upper bound τ = πM of proper time, but an arbitrary path will reach the singularity quicker, so using engines is advantageous.
A natural question, then, is how much time can we gain depending on the power of our spacecraft’s engines and the initial conditions of our crossing the event horizon, past which there is no escape? The details and the mathematics required to fully answer this question can be found in my recent paper, which can be found on arxiv and is currently undergoing peer review: https://arxiv.org/abs/2405.03510
In the process of this research, I have used Mathematica to create visualisations of worldlines of such a spacecraft from numerical calculations. I have also implemented methods for calculation of proper time depending on initial parameters. This notebook presents those visualisations in closer detail than the paper.
A natural question, then, is how much time can we gain depending on the power of our spacecraft’s engines and the initial conditions of our crossing the event horizon, past which there is no escape? The details and the mathematics required to fully answer this question can be found in my recent paper, which can be found on arxiv and is currently undergoing peer review: https://arxiv.org/abs/2405.03510
In the process of this research, I have used Mathematica to create visualisations of worldlines of such a spacecraft from numerical calculations. I have also implemented methods for calculation of proper time depending on initial parameters. This notebook presents those visualisations in closer detail than the paper.
Setup
Setup
xAct setup
xAct setup
We first have to initialise xAct -- which we will use for our relativistic calculations -- and set it up to use the Schwarzschild chart. We also constrain variables to remain in the region under the event horizon.
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<<MaTeX`texStyle={FontFamily->"Latin Modern Roman",FontSize->16,FontColor->Black};
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<<xAct`xCoba`<<xAct`ShowTime1`;$RecursionLimit=Infinity;Off[General::spell];Off[NDSolve::ndsz];Off[InterpolatingFunction::dmval];Off[FindRoot::cvmit];Off[FindRoot::lstol];Off[Reduce::ratnz];$DefInfoQ=False;$CVVerbose=False;$xCobaCacheVerbose=False;$PrePrint=ScreenDollarIndices;$Post=ScreenDollarIndices;$CVSimplify=Together;$CVReplace=False;$CovDFormat="Postfix";
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Package xAct`xPerm` version 1.2.3, {2015,8,23}
CopyRight (C) 2003-2020, Jose M. Martin-Garcia, under the General Public License.
Connecting to external linux executable...
Connection established.
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Package xAct`xTensor` version 1.2.0, {2021,10,17}
CopyRight (C) 2002-2021, Jose M. Martin-Garcia, under the General Public License.
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Package xAct`xCoba` version 0.8.6, {2021,2,28}
CopyRight (C) 2005-2021, David Yllanes and Jose M. Martin-Garcia, under the General Public License.
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These packages come with ABSOLUTELY NO WARRANTY; for details type Disclaimer[]. This is free software, and you are welcome to redistribute it under certain conditions. See the General Public License for details.
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In[]:=
DefManifold[M4,4,{μ,ν,α,β,σ}]
In[]:=
DefConstantSymbol[M]
In[]:=
DefChart[schw,M4,Range[0,3],{t[],r[],θ[],ϕ[]},FormatBasis{"Partials","Differentials"},ChartColorRed]
In[]:=
$Assumptions=And[(t[]|r[]|θ[]|ϕ[])∈Reals,2M>r[]>0,0<=ϕ[],M>0];
In[]:=
gschw=CTensorDiagonalMatrix-1-,,,,{-schw,-schw},0
2M
r[]
-1
1-
2M
r[]
2
r[]
2
(r[]Sin[θ[]])
Out[]=
CTensor-1+,0,0,0,0,,0,0,{0,0,,0},{0,0,0,},{-schw,-schw},0
2M
r
1
1-
2M
r
2
r
2
r
2
Sin[θ]
In[]:=
SetCMetric[gschw,schw,SignatureOfMetric->{3,1,0}]
Having set up the chart, we go straight to computing the necessary geometric tensors needed for calculations (Riemann tensor, Ricci tensor, and the Christoffel symbols for the covariant derivative).
In[]:=
CDschw=CovDOfMetric[gschw]//FullSimplify;
In[]:=
MetricCompute[gschw,schw,All]//AbsoluteTiming
Out[]=
{0.048032,Null}
In[]:=
Christoffel[CDschw,PDschw][α,-β,-δ]//FullSimplify
Out[]=
|
|
In[]:=
Riemann[CDschw][-α,-β,-δ,σ]//FullSimplify
Out[]=
In[]:=
Ricci[CDschw][-α,-β]//FullSimplify
Out[]=
0
Numerical calculations of geodesic equation
Numerical calculations of geodesic equation
Now that we’re done with defining the charts, it’s time to create some worldlines. The motion is restricted to the plane.
In[]:=
$Assumptions=And$Assumptions,θ[]==
π
2
Out[]=
(t|r|θ|ϕ)∈&&2M>r>0&&0≤ϕ&&M>0&&θ
π
2
In[]:=
DefParameter[λ]DefParameter[τ]DefConstantSymbol[amag]DefScalarFunction/@{γt,γr,γϕ,at,ar,aϕ,e,L};
In[]:=
x=γt[λ],γr[λ],,γϕ[λ]u=CTensor[D[x,λ],{schw},0]a=CTensor[{at[λ],ar[λ],0,aϕ[λ]},{schw},0]
π
2
Out[]=
γt[λ],γr[λ],,γϕ[λ]
π
2
Out[]=
CTensor[{[λ],[λ],0,[λ]},{schw},0]
′
γt
′
γr
′
γϕ
Out[]=
CTensor[{at[λ],ar[λ],0,aϕ[λ]},{schw},0]
We can also define our functions representing individual constants.
In[]:=
constants=Flatten@{Solve[u[-ν]CTensor[{0,0,0,1},{schw},0][ν]==L[λ],γϕ'[λ]],Solve[u[-ν]CTensor[{1,0,0,0},{schw},0][ν]==e[λ]//FullSimplify,γt'[λ]]}
Out[]=
[λ],[λ]-
′
γϕ
L[λ]
2
r
′
γt
e[λ]r
-2M+r
In[]:=
u[μ]/.constants//FullSimplify
Out[]=
|
|
In any worldline calculation, we want to use constants of motion to remove degrees of freedom. In this case, since our constants e (energy per unit mass) and L (angular momentum per unit mass) will be subject to change due to thrust applied by rocket engines, we cannot take them as constant. However, we still have the normalisation of the 4-velocity; the normalisation of the 4-acceleration; and the orthogonality of the 4-velocity to 4-acceleration.
In[]:=
unorm=First@Simplify@Solve[u[α]u[-α]==-1,γr'[λ]]anorm=Simplify@Solvea[α]a[-α]==,at[λ]aunorm=First@Simplify@Solve[a[α]u[-α]==0,ar[λ]]
2
amag
Out[]=
[λ]-
′
γr
(2M-r)r+(2M-r)[λ]+[λ]
2
′
γt
3
r
2
′
γϕ
r
Out[]=
at[λ],at[λ]
rr+(2M-r)-
2
ar[λ]
2
amag
2
aϕ[λ]
2
r
-2M+r
rr+(2M-r)-
2
ar[λ]
2
amag
2
aϕ[λ]
2
r
2M-r
Out[]=
ar[λ][λ]
(2M-r)at[λ](2M-r)[λ]+aϕ[λ][λ]
′
γt
3
r
′
γϕ
2
r
′
γr
The important part is played by the equation of accelerated motion:
Auxiliary functions for display and calculation
Auxiliary functions for display and calculation
We will need functions for prettifying the display of our graphs:
As well as a solver for our equation of motion that can return the interpolated functions representing the solution:
And, to round it off, a generator of parametric plots from a given solution:
As we will see, overshooting with our engines is a problem. To rectify that, we need a solver that stops when optimal trajectory is reached:
Geodesics
Geodesics
Accelerated motion graphs
Accelerated motion graphs
Let’s see how acceleration changes the outcome:
This example for radial geodesics illustrates that we can ‘overshoot’ when we apply too much acceleration, as seen in the bottom-most worldline. Same situation for angular momentum:
So, let’s stop accelerating the moment we kill our velocity and fall on the optimal geodesic:
Direct calculation of proper time depending on circumstances
Direct calculation of proper time depending on circumstances
Drawing orbits is a great starting point and a good visualisation, but we’re interested in some qualitative answers.
The proper time on a geodesic is described by the following integral:
The proper time on a geodesic is described by the following integral:
What we need is expressions for e and L depending on r; it turns out these are possible to derive and in the case of purely radial infall, we can obtain an analytic expression in terms of Carlson integrals:
In the case of infall with e = 0, we do not have an analytic expression but can nevertheless obtain a simple numerical integral:
In the general case, we have to use an explicit quadrature and adjust at every step as the equations have a single degree of freedom corresponding to the way we control our spacecraft and thus don’t admit explicit solutions. The optimal way to thrust turns out to be always opposite to the 3-velocity in the frame of an optimally falling observer with e = L = 0.
Potential gains
Potential gains
Equipped with the ways to calculate proper time directly, we could ask ourselves: how much can we really gain? For this purpose, we will use some realistic values for acceleration (on the order of g to 1000g; see paper for longer discussion), and the black hole masses for Sagittarius A*, Messier 87*, and Phoenix A* (the largest black hole found in nature). We use geometric units:
For radial infall, for parameters e = 1 (free fall from far away from the black hole) and e = 10 (colliding with the black hole at high relativistic speeds):
In the most extreme case, we more than triple our survival time!
Radial infall gains
Radial infall gains
Using these, we can draw some graphs of proper time depending on parameters:
We also want to see the graphs for the gain function:
Infall with e = 0 gains
Infall with e = 0 gains
Similarly, we can do this for the situation of angular momentum only:
General case gains
General case gains
Conclusion
Conclusion
The presented visualisations, while a very nice way to understand a little about this problem of extending survival time under the event horizon of a black hole, are only one part. I strongly recommend checking out the paper for a more indepth look: https://arxiv.org/abs/2405.03510
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK