Spaghettification

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The tidal forces acting on a spherical body in a gravitational field. The depicted field originates just below the diagram.

In astrophysics, spaghettification is the stretching of objects into long thin shapes (rather like spaghetti) in a very strong gravity field, and is caused by extreme tidal forces. In the most extremes cases, near black holes, the stretching is so powerful that no object can withstand it, no matter how strong its components are.

The word spaghettification comes from an example given by Stephen Hawking in his book A Brief History of Time, where he describes the plight of a fictional astronaut who, passing within a black hole's event horizon, is "stretched like spaghetti" by the gravitational gradient (difference in strength) from head to toe.

1 A simple example
2 Examples of weak and strong spaghettification
3 Why spaghettification is so strong near black holes
4 Simplified mathematical explanation
5 References

[edit] A simple example

Spaghettification is caused by the differences in gravitational forces acting on the four objects. Each object follows a slightly different path.
See also an animated version.

To understand this process, consider the diagram on the left. Four objects are allowed to fall towards a large mass, such as a planet or a star. Each object accelerates "straight down", towards the center of the planet. Widely spaced objects will therefore follow trajectories that converge. This causes the left hand and right hand objects to squash together. On the other hand, the bottom object falls faster than the top one, because the force of gravity decreases with distance. The object nearer the planet is pulled harder than the one that is further away, and so the top and bottom objects are pulled further apart. The net result of these two sets of effects is to distort the diamond shape into a longer and thinner form. (See the animation.) Now imagine the green blobs in the diagram are parts of a larger object. A rigid object will resist distortion, because internal forces will act against the tidal forces. However, whenever the tidal forces become large enough to overcome these internal forces, a rigid body will be forced to stretch.

[edit] Examples of weak and strong spaghettification

Objects falling towards an ordinary massive body will hit the surface before the tidal forces get strong enough to overcome the internal forces holding a body together. For example if you jump off the ground, in theory the tidal forces are stretching you while you are in the air, but the effect is far too weak to measure.

If the massive body is a black hole, there is no surface to halt the fall. As objects fall into a black hole, the tidal forces become stronger and stronger until nothing can resist them, so the infalling bodies are stretched into thin strings of matter. Eventually, close to the singularity, the tidal forces become strong enough to tear molecules apart. For this reason, it would be impossible for a human to survive entering a singularity. The point at which these tidal forces become fatal depends on the size of the black hole. For a very large black hole, such as those found at the center of galaxies, this point will lie well inside the event horizon, so an astronaut may cross the event horizon without noticing any squashing and pulling whatsoever (although it's only a matter of time, because once inside an event horizon, it is impossible to avoid falling towards the center). For small black holes whose Schwarzschild radius is much closer to the singularity, the tidal effects may become fatal before the astronaut even reaches the event horizon.

[edit] Why spaghettification is so strong near black holes

The reason for the difference in the strength of tidal forces near planets and near black holes becomes clear if you consider what determines their strength. Newton's Law of Gravitation says that gravitational force is directly proportional to the mass of the objects and inversely proportional to the square of the distance between their centers. In other words:

if you increase one of the masses by a factor of x, you increase the gravitational force by a factor of x. Black holes have very large masses, for example the smallest black hole that can be formed by natural processes has over twice the mass of the sun.
if you halve the distance between the masses, you quadruple the gravitational force; and if you reduce the distance by a factor of 10, you increase the force by a factor of 100.

Black holes have so much mass concentrated into a very small radius that the gravitational force near them is enormous. Also, since a black hole has no solid surface, it is possible to get so close that the distance between the nearest and furthest parts of the falling object is a significant percentage of the distance between the center of the black hole and the nearest part of the falling object. These two factors make the difference in gravitational pull between the nearest and furthest parts of the falling object sufficient to cause spaghettification.

[edit] Simplified mathematical explanation

This will show that the tidal force which causes spaghettification increases if:

the large mass increases.
the length of the falling object in the direction of its fall increases. In other words, tidal forces are a bigger problem for large objects than for small ones.
the falling object is closer to the large mass.

Thorough and precise proof of the last two items requires differential calculus. But high school algebra is enough to explain the principles.

Newton's Law of Gravitation says that

$F = G \frac{m_1 m_2}{r^2}$

where:

F is the strength of the gravitational force between two objects.
G is the gravitational constant.
m₁ is the mass of the first object.
m₂ is the mass of the second object.
r is the distance between the two objects.

Now let's calculate the difference between the gravitational forces on the parts of a falling object which are nearest to and furthest from a black hole. That means we have to calculate the gravitational force on each of the two parts, so we need to define labels for all the variables in the calculation. For simplicity we'll imagine that these two parts of the falling object have equal size and mass.

Label	What it represents
m_p	mass of the nearest and furthest part of the falling object (they are equal)
m₁	mass of the body (black hole) towards which the falling object is falling.
r_n	distance of the nearest part of the falling object from the black hole
r_{n + x}	distance of the furthest part of the falling object from the black hole, where x is the object's length in the direction pointing towards the black hole.
F_n	gravitational force on the nearest part of the falling object
F_f	gravitational force on the furthest part of the falling object

Now we make two copies of of Newton's equation, one for each part of the falling object:

$F_n = G \frac{m_1 m_p}{{r_n}^2}$

$F_f = G \frac{m_1 m_p}{(r_n + x)^2}$

The difference between the forces on the nearest and furthest parts is:

$F_n - F_f = G \frac{m_1 m_p}{{r_n}^2} - G \frac{m_1 m_p}{(r_n + x)^2}$

which we can rearrange into the more useful form:

$F_n - F_f = {G m_1 m_p} ( \frac{1}{{r_n}^2} - \frac{1}{(r_n + x)^2} )$

So the difference in force, i.e. the tidal force, becomes greater:

If the large mass m₁ increases. Black holes have enormous masses, so the tidal force is very strong.
If the value of $(\frac{1}{{r_n}^2} - \frac{1}{(r_n + x)^2})$ increases.

It's obvious that $(\frac{1}{{r_n}^2} - \frac{1}{(r_n + x)^2})$ increases if x increases, because an increase in x makes $\frac{1}{(r_n + x)^2}$ smaller. In other words, the tidal force increases as the length of the falling body increases.

It's almost as obvious that $(\frac{1}{{r_n}^2} - \frac{1}{(r_n + x)^2})$ increases if r_n decreases. In the most extreme case, where r_n is zero (when the nearest part of the object hits the singularity), the value of $\frac{1}{{r_n}^2}$ becomes infinite while the value of $\frac{1}{(r_n + x)^2}$ is still finite. So the tidal force which causes spaghettification gets stronger very rapidly as the object falls closer to the singularity.