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Deriving the Schwarzchild metric

The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. Of the solutions to the Einstein field equations, it is considered by some to be one of the simplest and most useful. As a result of this, some textbooks omit the rigorous derivation of this metric, provided below.

Assumptions and notation

Working in a coordinate chart with coordinates \left(r,\theta ,\phi ,t\right) labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows

  1. A spherically symmetric spacetime is one that is invariant under rotations and taking the mirror image.
  2. A static spacetime is one in which all metric components are independent of the time coordinate t (so that{\tfrac  \partial {\partial t}}g_{{\mu \nu }}=0) and the geometry of the spacetime is unchanged under a time-reversal t\rightarrow -t.
  3. A vacuum solution is one that satisfies the equation T_{{ab}}=0. From the Einstein field equations (with zero cosmological constant), this implies that R_{{ab}}=0 since contracting R_{{ab}}-{\tfrac  {R}{2}}g_{{ab}}=0 yields R = 0.
  4. Metric signature used here is (+,+,+,−).

Diagonalising the metric

The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, (r,\theta ,\phi ,t)\rightarrow (r,\theta ,\phi ,-t), all metric components should remain the same. The metric components g_{{\mu 4}} \mu \neq 4) change under this transformation as:

g_{{\mu 4}}'={\frac  {\partial x^{{\alpha }}}{\partial x^{{'\mu }}}}{\frac  {\partial x^{{\beta }}}{\partial x^{{'4}}}}g_{{\alpha \beta }}=-g_{{\mu 4}}
\mu \neq 4)

But, as we expect g'_{{\mu 4}}=g_{{\mu 4}} (metric components remain the same), this means that:

g_{{\mu 4}}=\,0 \mu \neq 4)

Similarly, the coordinate transformations(r,\theta ,\phi ,t)\rightarrow (r,\theta ,-\phi ,t) and (r,\theta ,\phi ,t)\rightarrow (r,-\theta ,\phi ,t) respectively give:

g_{{\mu 3}}=\,0 \mu \neq 3)
g_{{\mu 2}}=\,0 \mu \neq 2)

Putting all these together gives:

g_{{\mu \nu }}=\,0 \mu \neq \nu )

and hence the metric must be of the form:

ds^{2}=\,g_{{11}}\,dr^{2}+g_{{22}}\,d\theta ^{2}+g_{{33}}\,d\phi ^{2}+g_{{44}}\,dt^{2}

where the four metric components are independent of the time coordinate t (by the static assumption).

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Asteroid Impact Avoidance

Asteroid impact avoidance comprises a number of methods by which near-Earth objects (NEO) could be diverted, preventing destructive impact events. A sufficiently large impact by an asteroid or other NEOs would cause, depending on its impact location, massive tsunamis, multiple firestorms and an impact winter caused by the sunlight-blocking effect of placing large quantities of pulverized rock dust, and other debris, into the stratosphere.

A collision between the Earth and an approximately 10-kilometre-wide object 66 million years ago is thought to have produced the Chicxulub Crater and the Cretaceous–Paleogene extinction event, widely held responsible for the extinction of the dinosaurs.

While the chances of a major collision are not great in the near term, there is a high probability that one will happen eventually unless defensive actions are taken. Recent astronomical events—such as the Shoemaker-Levy 9 impacts on Jupiter and the 2013 Chelyabinsk meteor along with the growing number of objects on the Sentry Risk Table—have drawn renewed attention to such threats. NASA warns that the earth is unprepared.

Deflection efforts

Most deflection efforts for a large object require from a year to decades of warning, allowing time to prepare and carry out a collision avoidance project, as no known planetary defense hardware has yet been developed. It has been estimated that a velocity change of just 3.5/t × 10−2 m·s−1 (where t is the number of years until potential impact) is needed to successfully deflect a body on a direct collision trajectory. In addition, under certain circumstances, much smaller velocity changes are needed.[2] For example, it was estimated there was a high chance of 99942 Apophis swinging by Earth in 2029 with a 10−4 probability of passing through a ‘keyhole’ and returning on an impact trajectory in 2035 or 2036. It was then determined that a deflection from this potential return trajectory, several years before the swing-by, could be achieved with a velocity change on the order of 10−6 ms−1.[3]

An impact by a 10 kilometres (6.2 mi) asteroid on the Earth has historically caused an extinction-level event due to catastrophic damage to the biosphere. There is also the threat from comets coming into the inner Solar System. The impact speed of a long-period comet would likely be several times greater than that of a near-Earth asteroid, making its impact much more destructive; in addition, the warning time is unlikely to be more than a few months.[4] Impacts from objects as small as 50 metres (160 ft) in diameter, which are far more common, are historically extremely destructive regionally.

 

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Double university scholarship for former HSP student

Double university scholarship for former HSP student

Congratulations to EQI graduate Trinh Manh Do, who was recently awarded a prestigious Queensland University of Technology (QUT) Vice-Chancellor’s Scholarship to study a double degree in Maths and Science, majoring in Astrophysics.

The Vice-Chancellor’s program is QUT’s premier scholarship for students with outstanding academic achievement. Trinh, who studied High School Preparation (HSP) at Mitchelton State High School (SHS) and m at North Lakes State College, also received a QUT Dean’s Award for graduating from high school.

When he arrived in Australia from Vietnam in mid-2013, Trinh spoke limited English. His EQI academic journey began with the HSP to prepare for mainstream high school study and he progressed very quickly to excel in his studies. In early 2015, his particular interest in Physics and Astrophysics inspired us to make a short film about his school experiences.

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