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I am studying Noether’s theorem in field theory and I am not understanding what spacelike-surfaces mean. I will reproduce the bit of the argument below that contains the term “spacelike-sufaces” in the context I am not understanding.

There will be a conserved ccurrent for each group generator $a$. Each will result in a conserved charge (that is, an integral of motion). To see this, take in spacetime a volume unbounded in the space-like direction, but limited in time by two space-like surfaces $w_1$ and $w_2$. Integrating $\partial_{\mu} J^{\mu}_a=0$ over this volume, we get an integral over the boundary surface, composed of $w_1$, $w_2$ and the time-like boundaries supposed to be at infinity. If we now suppose the current to be zero at infinity on these boundaries, we remain with.

$$\int_{w_1}d\sigma_{\mu} J^{\mu}_a=\int_{w_2}d\sigma_{\mu} J^{

The hints pointing to two galaxies are found in the question mark’s strange shape. There are two brighter spots, one in the curve and the other in the dot, which could be the galactic nuclei, or the centers of the galaxies, Britt says. The curve of the question mark might be the “tails” being stripped off as the two galaxies spiral toward each other.

“It’s very cute. It’s a question mark … But you can find the colons and semicolons, and any other punctuation mark, because you have 10,000 little smudges of light in each image taken every half hour,” says David Helfand, an astronomer at Columbia University. The sheer number of shining objects we find are bound to create some serendipitous images, and our brains have evolved to find those patterns, he says.

Astronomers have seen similar objects closer to home. Two merging galaxies captured by the Hubble Space Telescope in 2008 also look like a question mark, just turned 90 degrees.

NASA has reestablished connection with Voyager 2 after a tense two weeks of not hearing anything from the probe. On July 21st, the agency lost contact with Voyager 2 following a series of planned commands that mistakenly pointed it two degrees away from our planet. While it is scheduled to automatically reset its orientation on October 15th, it’s not surprising that NASA scientists didn’t just wait for that date to know whether the spacecraft is still running. Voyager 2 was launched way back in 1977, and it’s one of the only two probes sending us back valuable data on interstellar space.

For a few days after July 21st, NASA wasn’t even sure what the spacecraft’s condition was. It wasn’t until August 1st that multiple ground antennas from the Deep Space Network (DSN) were able to detect a carrier signal from the probe. A carrier signal is what a spacecraft uses to beam data back to the ground, but NASA said the one DSN detected was too weak to be able to transmit any information. Still, it was enough to confirm that Voyager 2 was still working and that it hadn’t deviated from its trajectory.

Instead of simply waiting for October, Voyager’s ground team decided to take action. They concocted a plan to “shout” a command to the spacecraft across over 12.3 billion miles of space using the DSN, telling it to turn its antenna back to Earth. The whole process illustrated just how vast outer space truly is: It took 18.5 hours for that message to reach the probe, and another 18.5 hours for NASA to start receiving science and telemetry data again, indicating that Voyager 2 had received the command.