Concerning restricted access to the Temple Mount: It depends. For non-Muslims, it has to do with the religious tension that exists within the area. Christians and Jews, despite having some sort of religious connection to the area, aren't allowed to pray or make any sort of "religious demonstration"...
Best answer: Concerning restricted access to the Temple Mount: It depends. For non-Muslims, it has to do with the religious tension that exists within the area. Christians and Jews, despite having some sort of religious connection to the area, aren't allowed to pray or make any sort of "religious demonstration" there whatsoever, and have to restrict themselves to just being tourists. Apparently it is offensive to Muslims, even if it's outside the mosque and shrine areas, to show one's faith if it isn't Islamic. In addition, there is political pressure to keep Jews from the area as it'll be seen as a provocation. Basically, they really don't like the mostly Jewish Israelis, and think that increased Jewish visitation will for some reason will end up with them losing their rights or their holy spots. There is no precedent for believing such (quite the opposite) but it's part of what's preached up there and people just buy it.
However, when it comes to Muslim restrictions on the area: It has to do with security. I allude to the Imams riling up the masses in the previous paragraph, but here I'll directly state that they do encourage Palestinians (especially young, impressionable Palestinian men) to act in a hostile manner towards Israel. This includes rioting, which can get really bad. For example, I know of one instance where Molotov cocktails were thrown. Because Israel doesn't want to deal with these things, they naturally do what they can to reduce it....and that includes placing restrictions on who can attend. It is always in response to violence, and is usually lifted if there is a relative calm afterwards.
As for gay marriage: It's actually legal in Israel, technically speaking. What isn't is performing it within Israel's boarders; however, it is recognized if done abroad and one can get the full legal benefits that come with marriage that way. The ceremony of marriage is restricted to certain religious courts, none of which are interested in performing these ceremonies. Those who may, for example the Reform Jewish congregations, don't have the authority to marry people in a legal manner. All they can do is do ceremonies that do not carry any legal weight, but for those people it's "in the eyes of G-d" anyway who doesn't care about man-made law like that. So, gay couples (like many straight Israeli couples) go abroad to complete their legal unions.
4 days ago