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I suspect that it is possible as long as time dilation occurs during the process of moving one end of the bridge at relativistic speeds as in the following scenario:

Imagine an Einstein-Rosen bridge that can at the very least allow photons to be sent through it. These photons would traverse the bridge at the speed of light but faster than a normal light beam traversing the actual path between the two ends. When you accelerate one end any signals received at the other end will be time dilated in the same way that any direct signal from the accelerated object will be time dilated. Let’s leave end 1 on Earth and put end 2 on a very fast spaceship. Also attach end 2 to a clock that emits two photons every second. One photon travels the normal path between end 1 and end 2 while the other photon travels through the bridge. An observer on Earth sees both photons arriving at the same time. Now the end2/clock system accelerates away from Earth to near the speed of light. The observer on Earth sees the signals from the direct path slow down to one photon per year but also sees the signals from end 1 slow down to one photon per year. The observer on Earth sees the two photons at the same time at first but then as the distance between the two ends grows, the direct path photons are delayed by time dilation and the delay of actually having to traverse the distance. The end2/clock system now decelerates to zero velocity 5 light years away. Now the observer has aged 5 years while any second observer traveling with the end2/clock system on the spaceship will not have aged. If the observer at end 1 on Earth sends a message encoded in photons through the bridge to the observer at end2 telling him to turn off the clock, the observer at end1 will stop seeing the photons traveling through the bridge but he will continue seeing the clock tick when looking down the direct path. In fact after 5 years the observer on Earth will see the spaceship is only half way there since from their perspective it would take 2.5 years for a spaceship traveling near the speed of light to get halfway and then another 2.5 years for light to travel the distance back. After 10 total years the observer on Earth would see the clock stop ticking on the direct path. The reason why this does not violate causality is because if the observer at end 2 travels back to the observer at end 1, each end of the bridge will still be at the same time (10 years from the starting point) and the observer won't be able to communicate with his future or his past.

Is that scenario consistent with relativity? Are there other ideas for ways of maintaining causality given an Einstein-Rosen bridge?