Is it possible to understand Einstein’s general theory of relativity intuitively? As I understand it, all the pictures depicting how space-time causes the effects of gravity are completely wrong. 
The student can accept general relativity is correct because it correctly predicts real world phenomenon, but it is not actually possible to create a physical model of how general relativity works.
Am I correct in saying that a student has to trust the complex mathematics of general relativity and ignore his intuition?
 A: To answer your question, to intuitively understand GR, you have to start by accepting that it is not mass that bends spacetime, but stress-energy.


*

*The first thing to understand intuitively can be GR time dilation. This is understandable intuitively from the twin paradox.


Then you have to learn about the four speed vector, and that its magnitude has to be c.
Once you do that, you will realize that the four vector (x,y,z,t) will give you an answer why an object, without initial momentum, placed inside a gravitational field, will start moving in space along a geodesic towards the center of mass.
The object is moving in the time dimension always, but initially it is not moving in space. If you place it into a gravitational field, the effect of the gravitational field will be that it will slow down the object in the time dimension.
So its four vector's time component will change. Since the four vector's magnitude has to be c, the spatial components will have to compromise, and the spatial components of the four vector will start changing, the object will start moving in space, along a geodesic, towards the center of mass.
The way to understand it intuitively, is that the object is always moving in time, even if in space (relative to the center of mass) it is not moving. The gravitational field will slow it down in the time dimension.
The easiest way to understand this is to learn the twin paradox in GR way. In this case the twin on earth and the twin on the spaceship will not age differently as long as the spaceship is moving with constant speed. Because speed is symmetrically relative, the twin in the spaceship could say that the twin on earth is moving away, and the twin on earth could say the same. 
The difference comes when the spaceship turns around, and accelerates/decelerates. Again, its spatial components of its four vector will start changing, because of the acceleration, deceleration. This effect is the same as a gravitational field. Since it's spatial components of the four vector start changing, and the magnitude of the four vector has to be c, the time component of the four vector will compensate, the object will slow down in the time dimension.
The spaceship will slow down in the time dimension compared to the twin on earth. The twin in the spaceship will age slower then the twin on Earth until the spaceship is turning, accelerating. 
If the twins could see each other, the twin in the spaceship would see the twin on Earth age faster then himself. 
When the spaceship turned and travels backwards with constant speed, they do not age differently any more, but the twin on Earth is already older because of the turn. 
When they meet, the twin on Earth will be older.


*GR has another intuitively understandable effect, and that is bending spacetime. The Shapiro effect is a very good way to understand intuitively what happens when stress-energy bends spacetime. A photon traveling from a far away star, and traveling next to the sun will move along a bent spacetime, because the stress-energy of the sun will bend spacetime next to it.


As the photon moves next to the sun, time dilation, that was mentioned, will cause time to tick slower next to the sun (when viewed from Earth). So the clocks on Earth will tick faster. More time will pass on Earth.
Since spacetime is bent next to the sun, the photon will also have to travel a longer path (viewed from Earth).
So the speed of the photon will be distance/longer time = slower then speed c.
The speed of the photon will always be c when measured locally next to the sun.
It is only when we measure the speed of the photon next to the sun from  Earth that it will be slower then c. 
You can see this effect of bending spacetime when learning about gravitational lensing.
